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+{"text":"\\section{Introduction}\n\nIn recent years we have seen much progress in the field \nof network dynamics and dynamics on \nnetworks~\\cite{Albert2002,Newman2003,Boccaletti2006,Arenas2008}.\nStrong\ninterest in understanding phenomena such as disease spread on social networks, interaction online social media such as Facebook and Twitter,\ndynamics of neuronal networks, and many others have encouraged development of\nmathematical tools necessary to analyze the behavior of such \nsystems~\\cite{Goncalves2011,Josic2009,Sejnowski2001,moore2000,Pastor-Satorras2001}.\n\n\nOften the first step in analyzing such systems is to represent them as \nnetworks, where an individual unit, e.g., a person, a user account, a neuron, \nis represented by a node, and possibility of interaction between any two \nunits is represented by a link between them. The dynamical processes on such \nnetworks are often characterized by their statistical properties via a \nmean-field approach~\\cite{Gross2006b,Shaw2008a,Shkarayev2012,Jolad2011}. Such \nmean-field equations consist of a hierarchy of equations, where the expected \nstate of the nodes, due to interaction via the network, is coupled to the \nstatistical description of links in the network. The dynamical evolution of \nthe links in turn depends on the evolution of statistical description of node \ntriples, which in turn depend on higher order structures, and so on. In other \nwords, this mean-field description yields an infinite system of coupled \nequations, which usually must be truncated in order to be solvable. The \ntruncated system is open and has to be closed by introducing additional \ninformation about the system.\n\n\n\nThe dynamics on network systems are often closed at the \nlevel of link equations, where the network \ninformation makes its first \nappearance~\\cite{Murrell2004}. Perhaps the simplest \nclosure approach is based on the assumption of \nhomogeneous distribution of different node types in the \nsystem, and that the probability of finding a particular \ntype of node in the neighborhood of a given node is \nindependent of what else can be found in that node's \nneighborhood. This closure was shown to produce excellent \nresults for many different \nsystems~\\cite{Keeling1997,Murrell2004,Rogers2011,Taylor2012a,Kiss2012A,Kiss2012B}. \nThe \nheterogeneous mean-field approach, where conditioning on \nthe total degree of nodes is introduced, may improve the \naccuracy of the approximation, although drastically \nincreasing the number of equations in the \ndescription~\\cite{Marceau2010,Pastor-Satorras2001}. \nOften, additional information about the system, such as \nthe expected clustering coefficient, may be used to \nimprove the closure~\\cite{Taylor2012a}. In other cases, \nassumptions about the shape of degree distribution \nfunctions~\\cite{Kiss2012B}, possibly guided by \nnumerical simulations or physical \nobservations~\\cite{Keeling1997,Keeling2005}, may lead to \nan improvement in closure. \nEquation-free \napproaches may also be used when closing the mean-field \nequations~\\cite{Reppas2012,Gross2008}.\n\n\nAll of the above closures often lead to a reasonable \napproximation of the system dynamics. However, they all \nsuffer either from the lack of a priori knowledge of \nthe validity of approximation \nor from having an excessive number of equations \nthat must be analyzed. In this paper, we propose a new method that \nmay lead to accurate closures and that also allows one to manage the expectation of \nthe accuracy of the obtained closure. The proposed \napproach is based on simplification of the mean-field \nsystem of equations in some asymptotic regime. In the \nrest of the paper we demonstrate our approach by applying \nit to two adaptive network systems, i.e., networks where \ndynamical processes on the nodes affect the network \nstructure, which in turn affects subsequent dynamics on \nthe nodes \\cite{Gross2008a}. In section~\\ref{sec:NSR}, we derive a \nclosure for a system modeling recruitment to a \ncause~\\cite{Shkarayev2012}. In section~\\ref{sec:SIS} we \nderive an improved closure for an adaptive epidemic \nmodel~\\cite{Gross2006b}.\n\n\n\n \\begin{figure}[tb!]\n \\subfigure[]{\\includegraphics[]{Shkarfig1a}\\label{fig:nsr}}\n \\subfigure[]{\\includegraphics[]{Shkarfig1b}\\label{fig:nsr_rew}}\n \\hspace{10pt}\\subfigure[]{\\includegraphics[]{Shkarfig1c}\\label{fig:nsr_trp}}\n \\caption{\\subref{fig:nsr} Schematic representation of node dynamics in the recruitment model. The nodes are born into the N-class at rate $\\mu$; nodes die at rate\n$\\d$; the possible transitions between the classes are marked by the arrows and\nlabeled with the corresponding rates ($\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma$, $\\lB$, $\\g$). \\subref{fig:nsr_rew} Link rewiring takes place at a rate $w$.\n\\subref{fig:nsr_trp} Example of a node triple.\n}\\label{schem1}\n \\end{figure}\n\n\\section{Adaptive Recruitment model} \\label{sec:NSR}\n Our first example is a model for recruitment to a cause, \nintroduced in~\\cite{Shkarayev2012}. A society is modeled \nas a network in which some of its individuals represent a \nparticular ideology and actively recruit new members. \nThese nodes are referred to as the recruiting nodes, or \nR-nodes. The rest of the people in the society are either \nsusceptible to recruitment or non-susceptible, referred \nto as S- and N-nodes respectively. The N-nodes may \nspontaneously change their state and become S-nodes, and \nvice versa, at rates $\\lambda_1$ and $\\lambda_2$, \nrespectively. The R-nodes recruit S-nodes at a \nrecruitment rate $\\g$ per contact with S-node. A \nschematic representation of these transitions appears in \nFig.~\\ref{fig:nsr}. The R-nodes can improve their \nrecruiting capability by abandoning their connections to \nN-nodes in favor of S-nodes, as shown in \nFig~\\ref{fig:nsr_rew}. This rewiring process takes place \nat a rate $w$ per contact between R- and N-nodes. The \nsystem is open in the sense that nodes die at rate $\\d$ \nper node, and new nodes enter the system at rate $\\mu$. \nThe newborn nodes are born as N-nodes, and attach \nthemselves with links to $\\sigma$ \nrandomly chosen nodes.\n\n In order to describe the evolution of this system, we \nbegin with developing a heterogeneous mean-field \ndescription~\\cite{Marceau2010}. We characterize \nthe time evolution of $\\rho_{\\alpha;\\k}$, the expected \nnumber of nodes of type $\\alpha$ with $k_1$, $k_2$, and \n$k_3$ neighbors of type N, S, and R, respectively, in \ntheir neighborhoods, where $\\k=(k_1,k_2,k_3)$:\n \\begin{subequations}\n \\begin{align}\n \\begin{split}\n &\\dt \\rho_{N;{\\bf k}}} \\def\\Mnr{\\rho_{N;{\\bf k}-r_i} =\\\\\n &\\lB \\Ms-\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma \\rho_{N;{\\bf k}}} \\def\\Mnr{\\rho_{N;{\\bf k}-r_i} - \\d \\rho_{N;{\\bf k}}} \\def\\Mnr{\\rho_{N;{\\bf k}-r_i}+\\mu \\delta_{k_1+k_2+k_3,\\sigma}\\\\\n &+\\sum_{i}\\left[ \\wnr(r_i) \\Mnr- \\Omega_{N;\\k}} \\def\\wnr{\\Omega_{N;\\k-r_i}(r_i) \\rho_{N;{\\bf k}}} \\def\\Mnr{\\rho_{N;{\\bf k}-r_i}\\right],\n \\end{split}\\\\\n \\begin{split}\n &\\dt \\Ms =\\\\\n &-\\lB \\Ms-\\g k_3 \\Ms+\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma \\rho_{N;{\\bf k}}} \\def\\Mnr{\\rho_{N;{\\bf k}-r_i} - \\d \\Ms+\\\\\n &+\\sum_{i}\\left[ \\wsr(r_i) \\Msr- \\Omega_{S;\\k}} \\def\\wsr{\\Omega_{S;\\k-r_i}(r_i) \\Ms \\right].\n \\end{split}\\\\\n \\begin{split}\n &\\dt \\rho_{R;{\\bf k}}} \\def\\Mrr{\\rho_{R;{\\bf k}-r_i} =\\g k_3 \\Ms-\\d \\rho_{R;{\\bf k}}} \\def\\Mrr{\\rho_{R;{\\bf k}-r_i}+\\\\\n &+\\sum_{i}\\left[ \\wsr(r_i) \\Msr- \\Omega_{S;\\k}} \\def\\wsr{\\Omega_{S;\\k-r_i}(r_i) \\Ms \\right].\n \\end{split}\n \\end{align}\\label{eq:master_RSR}\n \\end{subequations} The allowed transitions and the \ncorresponding rates shown in the Table~\\ref{table:rsr}. \nIn the recruiting transition rates listed in the \ntable, function $P$ (function $Q$) corresponds to the \nexpected number of node chains that originate \nat a given N-node (S-node) with a neighborhood \nspecified by $\\k$ that is connected to an S-node, which \nis turn is connected to an R-node. The terms \n$N_{\\text{X}_1\\ldots \\text{X}_n}$ correspond to the \nexpected number of node chains in the system, where a \nnode chain constitutes a set of nodes, connected as \nfollows: a node of type $X_1$ is connected to the node of \ntype $X_2$, which in turn is connected to node of type \n$X_3$ etc. \nFor example, \n$\\s$ is the expected number of S-nodes in the network, \nwhile $\\rs$ is the expected number of links with S- and \nR-nodes at its ends. In our definition of node chains we \nrequire the $i$th and $i+1$st nodes to be different; \nhowever, $i$th and $i+2$nd nodes can in fact be \nthe same node. In the example of a network presented in \nFig.~\\ref{fig:nsr_trp} there are 4 RSR triples, \ncorresponding to the following node combinations: 1-2-1, \n1-2-3, 3-2-1, 3-2-3. Note that the order in which nodes \nappear matters, which, for example, means that \n$\\ss$ corresponds to the twice the expected number of \nundirected links between two susceptible nodes. \n\n\\begin{table}\n\\caption{\nTransitions and nonzero transition rates in} Eq.~(\\ref{eq:master_RSR})\n\\centering\n\\begin{tabular}{l|l}\n\\hline\\hline\ntransition & rate\\\\\n\\hline\n$r_1=(-1,1,0)$ & $\\Omega_{N;\\k}} \\def\\wnr{\\Omega_{N;\\k-r_i}(r_1)=\\Omega_{S;\\k}} \\def\\wsr{\\Omega_{S;\\k-r_i}(r_1)=\\Omega_{R;\\k}} \\def\\wrr{\\Omega_{R;\\k-r_i}(r_1)=\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma k_1$\\\\\n$r_2=(1,-1,0)$ & $\\Omega_{N;\\k}} \\def\\wnr{\\Omega_{N;\\k-r_i}(r_2)=\\Omega_{S;\\k}} \\def\\wsr{\\Omega_{S;\\k-r_i}(r_2)=\\Omega_{R;\\k}} \\def\\wrr{\\Omega_{R;\\k-r_i}(r_2)=\\lB k_2$\\\\\n$r_3=(-1,0,0)$ & $\\Omega_{N;\\k}} \\def\\wnr{\\Omega_{N;\\k-r_i}(r_3)=\\Omega_{S;\\k}} \\def\\wsr{\\Omega_{S;\\k-r_i}(r_3)=\\Omega_{R;\\k}} \\def\\wrr{\\Omega_{R;\\k-r_i}(r_3)=\\d k_1$\\\\\n$r_4=(0,-1,0)$ & $\\Omega_{N;\\k}} \\def\\wnr{\\Omega_{N;\\k-r_i}(r_4)=\\Omega_{S;\\k}} \\def\\wsr{\\Omega_{S;\\k-r_i}(r_4)=\\Omega_{R;\\k}} \\def\\wrr{\\Omega_{R;\\k-r_i}(r_4)=\\d k_2$\\\\\n$r_5=(0,0,-1)$ & $\\Omega_{N;\\k}} \\def\\wnr{\\Omega_{N;\\k-r_i}(r_5)=\\Omega_{S;\\k}} \\def\\wsr{\\Omega_{S;\\k-r_i}(r_5)=\\Omega_{R;\\k}} \\def\\wrr{\\Omega_{R;\\k-r_i}(r_5)=\\d k_3$\\\\\n$r_6=(1,0,0)$ & $\\Omega_{N;\\k}} \\def\\wnr{\\Omega_{N;\\k-r_i}(r_6)=\\Omega_{S;\\k}} \\def\\wsr{\\Omega_{S;\\k-r_i}(r_6)=\\Omega_{R;\\k}} \\def\\wrr{\\Omega_{R;\\k-r_i}(r_6)=$\\\\\n\t\t& $=\\sigma \\mu\/(\\n+\\s+\\r)$\\\\\n$r_7=(0,-1,1)$ & $\\Omega_{N;\\k}} \\def\\wnr{\\Omega_{N;\\k-r_i}(r_7)=\\g P(\\k)$, $\\Omega_{S;\\k}} \\def\\wsr{\\Omega_{S;\\k-r_i}(r_7)=\\g Q(\\k)$\\\\\n$r_8=(0,0,-1)$ & $\\Omega_{N;\\k}} \\def\\wnr{\\Omega_{N;\\k-r_i}(r_8)=w k_3$\\\\\n$r_9=(0,0,1)$ & $\\Omega_{S;\\k}} \\def\\wsr{\\Omega_{S;\\k-r_i}(r_9)=w\\rn\/\\s$\\\\\n$r_{10}=(-1,1,0)$ & $\\Omega_{R;\\k}} \\def\\wrr{\\Omega_{R;\\k-r_i}(r_{10})=w\\rn\/\\s$\\\\\n\\hline\n\\end{tabular}\n\\label{table:rsr}\n\\end{table}\n\n\n\n The heterogeneous mean-field equations are high \ndimensional and, therefore, are extremely difficult to \nanalyze. A common way to analyze the dynamics of social \nnetworks is via lower dimensional mean-field \nequations. These can be generated by multiplying the \nheterogeneous mean-field equations by \n$k_1^{i_1}k_2^{i_2}k_3^{i_3}$ for some non-negative \ninteger values of $i_j$, and summing over $\\k$. Thus, the \nequations describing node dynamics are obtained by taking \n$i_1+i_2+i_3=0$, as given in \nEqs.~(\\ref{eq:node_N})-(\\ref{eq:node_R}) of Appendix~\\ref{app:NSR}, while the \ndescription of the link dynamics is obtained by taking \n$i_1+i_2+i_3=1$, as given in \nEqs.~(\\ref{eq:link_NN})-(\\ref{eq:link_RR}).  \n\n The hierarchy of equations generated in this manner must \nbe truncated in order to obtain a finite dimensional \ndescription of the system. Such truncation leaves the \nsystem open and in need of closure. For example, the \nsystem of node and link equations in~(\\ref{eq:NSR_MF}) \ncontains the terms $N_{\\text{NSR}}} \\def\\rnr{N_{\\text{RNR}}$, $\\ssr$ and $\\rsr$, which are \nhigher order structures.  The usual approach to closure \ncomes from the assumption of homogeneous distribution of \nthe R-nodes in the neighborhood of susceptible nodes, which leads \nto the following closure equations:\n \\begin{subequations}\n \\begin{align}\n &\\frac{N_\\text{XSR}}{\\s}=\\frac{N_{\\text{XS}}}{\\s}\\frac{\\rs}{\\s},\\label{eq:closure_XSR}\\\\\n &\\frac{\\rsr}{\\s}=\\left(\\frac{\\rs}{\\s}\\right)^2+\\frac{\\rs}{\\s},\\label{eq:closure_RSR_old}\n \\end{align}\n \\end{subequations}\n where we also assumed that the degree distribution of \nsusceptible nodes is Poisson. The details of these closures are presented in the \nAppendix~\\ref{app:closure}. These closures are ad hoc and \nmay fail to capture the system behavior accurately if, \nfor example, correlations are present. Here we develop an \napproach that derives the closure based on the system \nbehavior in some asymptotic regime. In particular, we \nderive the equations that describe the evolution of node \ntriples, take a steady-state relation and consider it in \nthe asymptotic regime, where we are able to close the \nequations. Finally, we numerically explore the \nperformance of the derived closures in \nparameter regimes outside of the asymptotic limit and \noutside of the steady-state.\n\n \\subsection{Closing of NSR and SSR terms}\n We develop a closure of the $N_{\\text{NSR}}} \\def\\rnr{N_{\\text{RNR}}$ and $\\ssr$ terms by considering the \nevolution of the expected number of node triples in the limit of \n$\\g,\\d,\\mu\/(\\n+\\s+\\r) \\ll w, \\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma, \\lB$. We consider the following expression:\n \\begin{align}\n \\begin{split}\n &\\sum_{\\k}\\left(\\frac{\\rs}{\\s}\\frac{\\dt\\Ms}{\\n}+\\frac{\\rn}{\\n}\n\\frac{\\dt\\rho_{N;{\\bf k}}} \\def\\Mnr{\\rho_{N;{\\bf k}-r_i}}{\\n}\\right)\\left(k_1 k_3 + k_2 k_3\\right).\n \\end{split}\n \\end{align}\n This relation is evaluated at the steady state and using\nEq.~(\\ref{eq:nsr_limit1}) and~(\\ref{eq:nsr_limit2}).\nAfter some algebraic manipulations described in Appendix~\\ref{app:NSRSSR},\nthe above relation leads to\n \\begin{align}\n \\begin{split}\n &\\frac{N_{\\text{NSR}}} \\def\\rnr{N_{\\text{RNR}}}{\\s}+\\frac{\\ssr}{\\s}=\\frac{\\sn}{\\s}\\frac{\\rs}{\\s}+\\frac{\\ss}{\\s}\\frac{\\rs}{\\s}\n \\end{split}\\label{eq:SSR_NSR_closure}\n \\end{align}\n a result that is consistent with but does not imply the closure in \nEqs.~(\\ref{eq:closure_XSR}) for $X=N$ and $S$.\n\n We compare the asymptotically derived result of \nEq.~(\\ref{eq:SSR_NSR_closure}) and the ad hoc closure for \nthe $N_{\\text{SSR}}$ term in Eq.~(\\ref{eq:closure_XSR}) \nwith the corresponding values measured in the Monte Carlo \nsimulations. Figure~\\ref{fig:NSR_SSR} presents the \nrelative error of the two closures\n \\begin{align}\n \\Delta = \\left|1-\\frac{\\text{approximation}}{\\text{exact value}} \\right| \\label{eq:delta}\n \\end{align}\n where simulation measurements are used as the exact \nvalue. We can see in Fig.~\\ref{fig:SSR_NSR_11} that the \nexpected number of NSR and SSR triples per susceptible \nnode, $N_{\\text{NSR}}} \\def\\rnr{N_{\\text{RNR}}\/\\s+\\ssr\/\\s$, is well approximated (error on the order of \nabout 1\\% or less) by the \nclosure of Eq.~(\\ref{eq:SSR_NSR_closure}) in the large $\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma$ and \n$\\lB$ limit, further improving as $w$ is increased.\nAccording \nto Fig.~\\ref{fig:SSR_NSR_9}, the closure of \n$N_{\\text{NSR}}} \\def\\rnr{N_{\\text{RNR}}\/\\s+\\ssr\/\\s$ continues to hold even in the parameter \nregime outside of the considered limit. As for the \nindividual closures, Fig.~\\ref{fig:SSR_11} shows that the \nclosure of $\\ssr\/\\s$ holds as well in the large $\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma$ and \n$\\lB$ regime. However, as we can see in \nFig.~\\ref{fig:SSR_9}, the closure \nfails for small $\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma$ and $\\lB$, especially as $\\g$ \nbecomes dominant. Since the closure of $\\ssr\/\\s+N_{\\text{NSR}}} \\def\\rnr{N_{\\text{RNR}}\/\\s$ \nis derived in the asymptotic regime, we expect it to \nbe accurate at least in that limit, while $\\ssr\/\\s$ \nclosure is still ad hoc, and, therefore, deviation from \nsimulations is not unexpected.\n\n \\begin{figure}[tb!]\n \\subfigure[$\\ssr+N_{\\text{NSR}}} \\def\\rnr{N_{\\text{RNR}}$ $\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{1}, \\lB=10^{2}$]{\\includegraphics[]{Shkarfig2a}\\label{fig:SSR_NSR_11}}\n \\subfigure[$\\ssr$  ~~$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{1},\\lB=10^{2}$]{\\includegraphics[]{Shkarfig2b}\\label{fig:SSR_11}}\n \\subfigure[$\\ssr+N_{\\text{NSR}}} \\def\\rnr{N_{\\text{RNR}}$ $\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{-1}, \\lB=10^{0}$]{\\includegraphics[]{Shkarfig2c}\\label{fig:SSR_NSR_9}}\n \\subfigure[$\\ssr$ $\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{-1}, \\lB=10^{0}$]{\\includegraphics[]{Shkarfig2d}\\label{fig:SSR_9}}\n \\caption{Relative errors in $N_{\\text{NSR}}} \\def\\rnr{N_{\\text{RNR}}+\\ssr$ closure \n(Eq.~(\\ref{eq:SSR_NSR_closure})) (panels a,c) and $\\ssr$ \nclosure (Eq.~(\\ref{eq:closure_XSR})) (panels b,d) \nat steady \nstate, as a function of $\\g$ for $w=10^{0}$ (circle, \ngreen online), $w=10^{1}$ (triangle, red online), \n$w=10^{2}$ (cross, black online). The simulations are \nperformed following the continuous time algorithm \nintroduced in~\\cite{Gillespie}. The other parameters are \n$\\d=1$, $\\sigma=10$, $\\mu=10^5$.\n}\\label{fig:NSR_SSR}\n \\end{figure}\n\n\n \\subsection{Closure of RSR term}\n In order to develop a closure for the $\\rsr$ term, we consider the expression\n \\begin{align}\n \\begin{split}\n &\\sum_{\\k}k_3^2\\dt\\Ms,\n \\end{split}\n \\end{align}\n which leads to the equation describing the evolution of \nthe expected number of RSR triples:\n \\begin{align}\n \\begin{split}\n &\\dt \\rsr = - \\lB \\rsr - \\g \\sum_{\\k} k_3^3 \\Ms (\\k) +\\\\\n & \\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma \\rnr-\\d \\rsr+\\g(2N_\\text{RSSR}+\\ssr)+\\\\\n &w \\frac{\\rn}{\\s} (2\\rs+\\s)+\\d(-2 \\rsr + \\rs).\\label{eq:rsr_time}\n \\end{split}\n \\end{align}\nAnalyzing the steady state of this equation in the limit where\n$\\g,\\d,\\mu\/(\\n+\\s+\\r) \\ll w, \\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma, \\lB$, and utilizing the relations in Eqs.~(\\ref{eq:nsr_limit1})\nand~(\\ref{eq:nsr_limit2}), we are able to solve this equation and obtain the\nfollowing relation:\n \\begin{align}\n \\frac{\\rsr}{\\s} = \\left(\\frac{\\rs}{\\s}-\\frac{\\rn}{\\n}\\right)\\left(2\\frac{\\rs}{\\s}+1\\right)+\\frac{\\rnr}{\\n}.\n\\label{eq:rsr}\n \\end{align}\n Note that, unlike the ad hoc closure of Eq.~(\\ref{eq:closure_RSR_old}), this result should at least be\naccurate in the considered limit, and, therefore should be more reliable.\n\n In order to close the $\\rnr\/\\n$ term in \nEq.~(\\ref{eq:rsr}) we analyze the limiting behavior of \nthe following expression:\n \\begin{align}\n \\begin{split}\n &\\sum_{\\k}k_3^2[\\dt\\Ms + \\dt\\rho_{N;{\\bf k}}} \\def\\Mnr{\\rho_{N;{\\bf k}-r_i}],\n \\end{split}\n \\end{align}\nwhich in steady state reduces to:  \n \\begin{align}\n &\\frac{\\rnr}{\\n}=\\frac{\\rn}{\\n}+\\frac{\\rn}{\\n}\\frac{\\rs}{\\s}\\label{eq:rnr_closure}\n \\end{align}\n Upon substituting the result of Eq.~(\\ref{eq:rnr_closure}) into Eq.~(\\ref{eq:rsr}) we obtain the following closure of the $\\rsr\/\\s$ term:\n \\begin{align}\n \\frac{\\rsr}{\\s} =\n2\\left(\\frac{\\rs}{\\s}\\right)^2+\\frac{\\rs}{\\s}-\n\\frac{\\rs}{\\s}\\frac{\\rn}{\\n}.\\label{eq:closure_RSR_new}\n \\end{align}\n\n\n In Fig.~\\ref{fig:NSR_RSR}, we compare the performance of \nthe new closure of $\\rsr$ in \nEq.~(\\ref{eq:closure_RSR_new}) to the ad hoc, homogeneity \nbased closure of Eq.~(\\ref{eq:closure_RSR_old}). We \nconsider the numerical solution of the mean-field \nequations, found in Appendix~\\ref{app:NSR}, closed \naccording to the two methods, and \ncompare those to the steady-state size of the \nrecruiting class measured in the direct network \nsimulations. In both cases, the $N_\\text{NSR}$ and \n$N_\\text{SSR}$ terms are \nclosed according to the homogeneity assumption in \nEq.~(\\ref{eq:closure_XSR}). Thus, in \nFig.~\\ref{fig:NSR_100_3} we see that in the considered \nlimit, i.e., when $\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma, \\lB$ and $w$ are large, the \nmean-field closed using our approach is in much better \nagreement with the simulations than the ad hoc assumption \nof Eq.~(\\ref{eq:closure_RSR_old}). The reason for the \nsuperior performance lies in the better approximation of \nthe $\\rsr\/\\s$ term, shown in \nFig.~\\ref{fig:NSR_RSR_100_3}. Here $\\rsr\/\\s$ and \n$\\rs\/\\s$ are parametrized by $\\g$, with larger values of \n$\\rs\/\\s$ corresponding to the larger values of $\\g$. \nNotice that the performance of the closure is reduced at \nthe larger values of $\\g$, as the system moves outside of \nthe considered limit.\n\n\n \\begin{figure}[tb!]\n \\subfigure[$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{1}, \\lB=10^{2}$ $w=10^2$]{\\includegraphics[width=1.54in]{Shkarfig3a}\\label{fig:NSR_100_3}}\n \\subfigure[$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{1}, \\lB=10^{2}$ $w=10^2$]{\\includegraphics[width=1.54in]{Shkarfig3b}\\label{fig:NSR_RSR_100_3}}\n \\subfigure[$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{1}, \\lB=10^{2}$ $w=10^{-1}$]{\\includegraphics[width=1.54in]{Shkarfig3c}\\label{fig:NSR_100_0}}\n \\subfigure[$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{1}, \\lB=10^{2}$ $w=10^{-1}$]{\\includegraphics[width=1.54in]{Shkarfig3d}\\label{fig:NSR_RSR_100_0}}\n \\subfigure[$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{-1}, \\lB=10^{0}$ $w=10^{2}$]{\\includegraphics[width=1.54in]{Shkarfig3e}\\label{fig:NSR_1_3}}\n \\subfigure[$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{-1}, \\lB=10^{0}$ $w=10^{2}$]{\\includegraphics[width=1.54in]{Shkarfig3f}\\label{fig:NSR_RSR_1_3}}\n \\subfigure[$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{-1}, \\lB=10^{0}$ $w=10^{-1}$]{\\includegraphics[width=1.54in]{Shkarfig3g}\\label{fig:NSR_1_0}}\n \\subfigure[$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^{-1}, \\lB=10^{0}$ $w=10^{-1}$]{\\includegraphics[width=1.54in]{Shkarfig3h}\\label{fig:NSR_RSR_1_0}}\n \\caption{Recruitment level, $\\r$, and expected number of \nRSR triples per S-node, $\\rsr\/\\s$, as a function of \nrecruitment rate $\\g$, for several sets of parameters \n$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma$, $\\lB$ and $w$. Simulation results are shown by \ncircles (red online). In \\subref{fig:NSR_100_3}, \n\\subref{fig:NSR_100_0}, \\subref{fig:NSR_1_3} \nand~\\subref{fig:NSR_1_0}, the curves correspond to \nsolution of mean-field equations, while in \n\\subref{fig:NSR_RSR_100_3}, \\subref{fig:NSR_RSR_100_0}, \n\\subref{fig:NSR_RSR_1_3} and~\\subref{fig:NSR_RSR_1_0} the \ncurves correspond to the approximation of $\\rsr\/\\s$ using \ntwo different closures. Dark curves (black online) \ncorrespond to closure in Eq.~(\\ref{eq:closure_RSR_new}), \nwhile light curves (green online) correspond to closure \nin Eq.~(\\ref{eq:closure_RSR_old}). The other parameters \nare same as in Fig.~\\ref{fig:NSR_SSR}. Note that in \nFig.~\\subref{fig:NSR_100_0},~\\subref{fig:NSR_RSR_100_0},~\\subref{fig:NSR_1_0}, \nand~\\subref{fig:NSR_RSR_1_0} the curves corresponding to \nthe two analytic solutions lie on top of each other.\n}\\label{fig:NSR_RSR}\n \\end{figure}\n\n\nThe appeal of this approach is evident when we test it outside of the \nderivation limit. In Figs.~\\ref{fig:NSR_100_0} and~\\ref{fig:NSR_RSR_100_0} we \nsee that, when we reduce $w$, the mean-field recruited fraction and the RSR \nclosure continue to be in a good agreement with the simulations. We also note \nthat in this limit the new closure approaches the homogeneity closure. We \nnote that when the rewiring is slow relative to transitions between N and S, \nthe expected number of R neighbors should be similar for the two node types. \nThis would make the last term in Eq.~(\\ref{eq:closure_RSR_new}) approach \n$(\\rs\/\\s)^2$, explaining why the two closures are close. In \nFig.~\\ref{fig:NSR_1_3}, as $\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma$ and $\\lB$ are reduced, the new \nmean-field solution appears to be less consistent with the simulations, which \nis also reflected in the closure in Fig.~\\ref{fig:NSR_RSR_1_3}. Finally, in \nFig.~\\ref{fig:NSR_1_0} all of the parameters are about the same order, and \nyet the asymptotically derived closure and the corresponding mean-field \nare very much consistent with the simulations.\n\n\n\nThus far we have shown that our method has produced a \nclosure that is a good match for the simulated system in \nsteady-state, and is either superior to or as good as the \nad hoc homogeneity closure. We further test the \nperformance of our closure by using it outside of \nsteady-state. Figures~\\ref{fig:NSR_time} \nand~\\ref{fig:NSR_time_triple} show that our closure \ncontinues to be consistent with the simulations even \nduring the transient period. This suggests that the time \nderivative of $\\rsr$ in Eq.~(\\ref{eq:rsr_time}) can\nbe neglected in the considered limit.\n\n \\begin{figure}[tb!]\n \\subfigure[]{\\includegraphics[]{Shkarfig4a}\\label{fig:NSR_time}}\n \\subfigure[]{\\includegraphics[]{Shkarfig4b}\\label{fig:NSR_time_triple}}\n \\caption{ Figure~\\subref{fig:NSR_time} contains \nmeasurement and approximation of $\\r$. The circles (red \nonline): simulation results, the light curve (green \nonline): mean-field with homogeneous closure, the dark \ncurve (black online): mean-field with asymptotically \ndeveloped closure from \nEq.~(\\ref{eq:closure_RSR_new}). \nFigure~\\subref{fig:sis_time_triple} shows the time \nevolution of the number of RSR triples per S-node \n(circles, red online), the approximate value obtained \nfrom the relation in Eq.~(\\ref{eq:closure_RSR_old}) \n(light curve, green online) and from \nEq.~(\\ref{eq:closure_RSR_new}) (dark curve, black \nonline). The simulations are performed with $w=10^2$, \n$\\lambda_1} \\def\\lB{\\lambda_2} \\def\\d{\\theta} \\def\\g{\\gamma=10^1$, $\\lB=10^2$ and $\\g=3.0$. The system evolves \nfrom a realization of Erd\\\"{o}s-R\\'{e}nyi network, with mean degree $10$ and \n$10^{5}$ nodes, 85{\\%} of which are N-nodes, 5\\% S-nodes and 10\\% \nR-nodes. The results are averaged over 10 dynamical \nrealizations.\n}\n \\end{figure}\n\n\n\n \\section{Adaptive Epidemic model}\\label{sec:SIS}\n The other example that we consider is a model for epidemic spread on an adaptive social\nnetwork~\\cite{Gross2006b}.  Here the disease spread is described using\nthe susceptible-infected-susceptible model, where each individual in the society\nis in one of the two states: sick or {\\it infected}, and healthy but {\\it\nsusceptible} to infection. In the framework of networks, we refer to these as I-\nand S-nodes respectively. The infected individuals become susceptible at recovery rate\n$r$. The disease can spread at a rate $p$ from infected individuals to\nsusceptible ones via a contact between them, where the existence of the contact\nis defined by the network structure. The adaptation mechanism allows\nsusceptible individuals to change their local connectivity to avoid\ncontact with infected individuals. Thus, the susceptibles rewire their contacts\naway from infecteds at rate $w$, connecting instead to a randomly chosen susceptible. The node and link\ndynamical rules are summarized in the Fig.~\\ref{fig:sis} and~\\ref{fig:sis_rew} respectively.\n\n \\begin{figure}[tb!]\n \\subfigure[]{\\includegraphics[]{Shkarfig5a}\\label{fig:sis}} \\hspace{20pt}\n \\subfigure[]{\\includegraphics[]{Shkarfig5b}\\label{fig:sis_rew}}\n \\caption{Schematic representation of \\subref{fig:sis} node dynamical rules and \\subref{fig:sis_rew} link\ndynamical rules in the adaptive epidemic model.\n}\\label{schem2}\n \\end{figure}\n\nThe evolution of the ensemble average of such a system is described by\nthe set of heterogeneous mean-field equations:\n \\begin{subequations}\n \\begin{align}\n \\begin{split}\n &\\dt {\\rho}_{S;\\kt}} \\def\\Mit{{\\rho}_{I;\\kt} = r\\Mit-p k_2 {\\rho}_{S;\\kt}} \\def\\Mit{{\\rho}_{I;\\kt}+\\\\\n &+\\sum_{i} \\left[\\wstr(r_i) {\\rho}_{S;\\kt-r_i}} \\def\\Mitr{{\\rho}_{I;\\kt-r_i}- \\wst(r_i) {\\rho}_{S;\\kt}} \\def\\Mit{{\\rho}_{I;\\kt}\\right],\n \\end{split}\\label{eq:masterS}\\\\\n \\begin{split}\n &\\dt \\Mit = -r\\Mit+p k_2 {\\rho}_{S;\\kt}} \\def\\Mit{{\\rho}_{I;\\kt}+\\\\\n &+\\sum_{i}\\left[ {\\Omega}_{I;\\kt-r_i}} \\def\\wstr{{\\Omega}_{S;\\kt-r_i}(r_i) \\Mitr- {\\Omega}_{I;\\kt}} \\def\\wst{{\\Omega}_{S;\\kt}(r_i) \\Mit\\right],\n \\end{split}\\label{eq:masterI}\n \\end{align}\\label{eq:master_SIS}\n \\end{subequations}\n where the value of ${\\rho}_{S;\\kt}} \\def\\Mit{{\\rho}_{I;\\kt}$ (value of $\\Mit$) corresponds \nto the number of S-nodes (I-nodes) with $k_1$ of S-nodes \nand $k_2$ of I-nodes in their neighborhoods, with \n$\\kt\\equiv (k_1,k_2)$. The function $K$ (function $M$) \ncorresponds to the expected number of node chains that \noriginate at an S-node (I-node) with a neighborhood \nspecified by $\\k$, which connects to an S-node, which in \nturn connects to an I-node. The functions \n$N_{\\text{X}_1\\ldots \\text{X}_n}$ are defined in the same \nway as in Section~\\ref{sec:NSR}.\n\n\n\\begin{table}\n\\caption{\nTransitions and nonzero transition rates in\nEq.~(\\ref{eq:master_SIS}).}\n\\centering\n\\begin{tabular}{l|l}\n\\hline\\hline\ntransitions & non-zero rates\\\\\n\\hline\n$r_1=(1,-1)$ & ${\\Omega}_{I;\\kt}} \\def\\wst{{\\Omega}_{S;\\kt}(r_1)=\\wst(r_1)=r k_2$ \\\\\n$r_2=(-1,1)$ & $\\wst(r_2)=p K(\\kt)$, ${\\Omega}_{I;\\kt}} \\def\\wst{{\\Omega}_{S;\\kt}(r_2)=p M(\\kt)$\\\\\n$r_3=(1,-1)$ & $\\wst(r_3)=w k_2$\\\\\n$r_4=(1,0)$ & $\\wst(r_4)=w \\is\/\\s$\\\\\n$r_5=(-1,0)$ & ${\\Omega}_{I;\\kt}} \\def\\wst{{\\Omega}_{S;\\kt}(r_5)=w k_1$\\\\\n\\hline\n\\end{tabular}\n\\label{table:sis}\n\\end{table}\n\n\n The mean-field equations are generated by multiplying \nthe heterogeneous mean-field equations \nby $k_1^{i_1}k_2^{i_2}$ and summing over $\\k$, where \n$i_1$ and $i_2$ are nonnegative integers. Thus, two node \nequations are generated for $i_1+i_2=0$, and three \ndistinct link equations are generated for $i_1+i_2=1$. \nThese equations, presented in the appendix as \nEqs.~(\\ref{eq:node_S2})-(\\ref{eq:node_II}), are open due \nto dependence on terms describing the expected \nnumber of ISI triples, $\\isi$, and SSI triples, $\\ssi$. \nIn order to close this system of equations, additional \ninformation is required. Once again, the usual \napproach~\\cite{Keeling1997} is to make an assumption \nthat infected nodes are homogeneously distributed in \nthe neighborhood of S-nodes, an assumption that leads to \nthe following closure:\n \\begin{subequations}\n \\begin{align}\n &\\frac{\\ssi}{\\s}=\\frac{\\ss}{\\s}\\frac{\\is}{\\s},\\label{eq:closure_SSI_old}\\\\\n &\\frac{\\isi}{\\s}=\\left(\\frac{\\is}{\\s}\\right)^2+\\frac{\\is}{\\s},\\label{eq:closure_ISI_old}\n \\end{align}\n \\end{subequations}\n where we make an additional assumption that the total \ndegree distribution of susceptible nodes is Poisson. \nMore details can be found in \nAppendix~\\ref{app:closure}.\n\n \\begin{figure}[tb!]\n \\includegraphics[]{Shkarfig6}\n \\caption{Schematic representation of four-node chains I-S-S-I. The term $N_\\text{ISSI}$ corresponds to the\nexpected number of such chains.}\\label{fig:issi}\n \\end{figure}\n\nWe derive a new closure of the ISI term by first considering \nthe evolution of the number of ISI triples. Multiplying \nequation~(\\ref{eq:masterS}) by $k_2^2$ and \nsumming over $\\k$ at steady state, \n \\begin{align}\n \\begin{split}\n \\sum_{\\k} k_2^2 \\dt{\\rho}_{S;\\kt}} \\def\\Mit{{\\rho}_{I;\\kt} = 0,\n \\end{split}\n \\end{align}\n we obtain the following equation:\n \\begin{align}\n \\begin{split}\n & 0=r \\iii - p \\sum_{\\k} [k_2^3 {\\rho}_{S;\\kt}} \\def\\Mit{{\\rho}_{I;\\kt}] +\\\\\n & +(r+w)[-2\\isi+\\is] +p [2N_\\text{ISSI}+\\ssi], \\label{eq:isi_time}\n \\end{split} \n \\end{align}\nwhere the four-point term ISSI corresponds to the total \nnumber of node configurations shown in \nFig.~\\ref{fig:issi}. Using the steady-state relations in \nEqs.~(\\ref{eq:ss1}) and~(\\ref{eq:ss2}), we arrive at\n \\begin{align}\n \\begin{split} \n&2(r+w)\\s\\left(\\frac{\\isi}{\\s}-\\frac{\\is}{\\s}-\\frac{\\is}{\\s}\\frac{N_\\text{ISSI}}{\\ssi}\\right)=\\\\ \n&=r \\i\\left(\\frac{\\iii}{\\i} - \\frac{\\ii}{\\i}\\frac{\\sum_{\\k} [k_2^3 {\\rho}_{S;\\kt}} \\def\\Mit{{\\rho}_{I;\\kt}]}{\\sum_{\\k} [k_2^2 {\\rho}_{S;\\kt}} \\def\\Mit{{\\rho}_{I;\\kt}] }\\right).\\label{eq:isi_first}\n \\end{split}\n \\end{align}\n The left hand side of the equation corresponds \nto the flux of the expected number of ISI \ntriples due to the changes in the neighborhood \nof the susceptible nodes, while the right hand \nside corresponds to the flux due to the \ninfection and recovery of the susceptible node in the ISI \ntriple. In the limit of large infection rate and \nweak rewiring, the amount of time any node \nspends in the susceptible state approaches zero. \nTherefore, it is reasonable to assume that the \nflux of triples due to the changes in the \nneighborhood of the susceptible node will \napproach zero as well. This leads us to conclude \nthat the two sides of Eq.~\\ref{eq:isi_first} \nmust vanish, leaving us with the following \nrelation:\n \\begin{align}\n \\frac{\\isi}{\\s}= \\frac{\\is}{\\s}+\\frac{\\is}{\\s}\\frac{N_\\text{ISSI}}{\\ssi}\\label{eq:isi}\n \\end{align}\nFinally, we note that the term $N_\\text{ISSI}\/\\ssi$ corresponds \nto the expected number of I-nodes, node 1 in \nFig.~\\ref{fig:issi}, attached to the chain of nodes \nnumbered 2, 3 and 4 in that figure. This relation is \nwell approximated by the homogeneity assumption, that the \ninformation about the neighborhood of the 3rd node in \nFig.~\\ref{fig:issi} has no effect on the information \nabout the neighborhood of the 2nd node. In other words, \nthe following moment closure is considered:\n \\begin{align}\n &\\frac{N_\\text{ISSI}}{\\ssi}=\\frac{\\ssi}{\\ss}.\n \\end{align}\n In other words, we make a homogeneity \nassumption about the neighborhood of a neighbor, \nand we expect this assumption to be more \naccurate than the same assumption about a given \nnode's neighborhood, i.e., the closures in \nEqs.~(\\ref{eq:closure_SSI_old}) \nand~(\\ref{eq:closure_ISI_old}).\n\n \\begin{figure}[tb!]\n \\subfigure[~$p\/r=10^{-1\/2}$]{\\includegraphics[]{Shkarfig8a}\\label{fig:ISI_W1}}\n \\subfigure[~$p\/r=10^{-1\/2}$]{\\includegraphics[]{Shkarfig8b}\\label{fig:ISI_WD1}}\n \\subfigure[~$p\/r=10^{0}$]{\\includegraphics[]{Shkarfig8c}\\label{fig:ISI_W2}}\n \\subfigure[~$p\/r=10^{0}$]{\\includegraphics[]{Shkarfig8d}\\label{fig:ISI_WD2}}\n \\subfigure[~$p\/r=10^{1}$]{\\includegraphics[]{Shkarfig8e}\\label{fig:ISI_W3}}\n \\subfigure[~$p\/r=10^{1}$]{\\includegraphics[]{Shkarfig8f}\\label{fig:ISI_WD3}}\n \\caption{Number of ISI triples per S-node as a function \nof rewiring rate, for several infection rates: \nsimulations compared to the moment closures of \nEq.~(\\ref{eq:closure_ISI_old}) and \nEq.~(\\ref{eq:closure_ISI_new}). \nFigures~\\subref{fig:ISI_W1}, \\subref{fig:ISI_W2}, \nand~\\subref{fig:ISI_W3} show $\\isi\/\\s$ measured in \nsimulation (circles, red online), approximated using \nhomogeneity closure of \nEq.~(\\ref{eq:closure_ISI_old}) (light curve, green \nonline), and approximated using the result of asymptotic \nanalysis in Eq.~(\\ref{eq:closure_ISI_new}) (dark curve, \nblack online). The closures are evaluated using \nnode and link quantities measured in the \nsimulations. Light curves (green online) in \nfigures~\\subref{fig:ISI_WD1}, \\subref{fig:ISI_WD2}, \nand~\\subref{fig:ISI_WD3} show the relative error, \nEq.~(\\ref{eq:delta}), of the homogeneity approximation, \nwhile dark curves (black online) show the relative error \ndue to the newly derived approximation. \nCusps in the relative error curves correspond to the \n$\\isi\/\\s$ homogeneity closure curve crossing through the \ncurve measured in simulations. Simulations are performed \non a network with $10^5$ nodes and $5\\times10^5$ links \nfollowing algorithm in~\\cite{Gillespie}.} \n\\label{fig:ISI_W}\n \\end{figure}\n\n \\begin{figure}[tb!]\n \\subfigure[$w\/r=10^{-1}$]{\\includegraphics[]{Shkarfig9a}\\label{fig:ISI_P1}}\n \\subfigure[$w\/r=10^{-1}$]{\\includegraphics[]{Shkarfig9b}\\label{fig:ISI_PD1}}\n \\subfigure[$w\/r=10^{0}$]{\\includegraphics[]{Shkarfig9c}\\label{fig:ISI_P2}}\n \\subfigure[$w\/r=10^{0}$]{\\includegraphics[]{Shkarfig9d}\\label{fig:ISI_PD2}}\n \\subfigure[$w\/r=10^{1}$]{\\includegraphics[]{Shkarfig9e}\\label{fig:ISI_P3}}\n \\subfigure[$w\/r=10^{1}$]{\\includegraphics[]{Shkarfig9f}\\label{fig:ISI_PD3}}\n \\caption{Number of ISI triples per S-node as a function of infection rate, for\nseveral rewiring rates: simulations compared to two approximations. The curves\nand circles are defined as in Fig.~\\ref{fig:ISI_W}, with the same network size\nand number of links.}\\label{fig:ISI_P}\n \\end{figure}\n\n\n\nThus, we have derived a new closure of $\\isi$: \n \\begin{align}\n &\\frac{\\isi}{\\s}=\n\\frac{\\is}{\\s}+\\frac{\\is}{\\s}\\frac{\\ssi}{\\ss},\\label{eq:closure_ISI_new}\n \\end{align}\nwhich relies on our ability to close the $\\ssi$ term, and this brings \nus one step closer to finding an accurate closure of the \nmean-field description of the adaptive epidemic \nmodel~(\\ref{eq:SIS_MF}). Curiously, the \nhomogeneity closure of $\\ssi$ in \nEq.~(\\ref{eq:closure_SSI_old}), together with \nEq.~(\\ref{eq:closure_ISI_new}), leads to the homogeneity \nclosure in Eq.~(\\ref{eq:closure_ISI_old}).  \nThus, as is suggested by Figs.~\\ref{fig:ISI_W} \nand~\\ref{fig:ISI_P}, where the measured values of \n$\\ssi$ are used, improving the closure of $\\ssi$ \nbeyond the homogeneity assumption leads to \nimprovement of the $\\isi$ closure. In fact, \nFigs~\\ref{fig:ISI_WD1},~\\ref{fig:ISI_WD2},~\\ref{fig:ISI_WD3} \nas well \nas~\\ref{fig:ISI_PD1},~\\ref{fig:ISI_PD2},~\\ref{fig:ISI_PD3} \nshow the relative deviation of the closure relations from \nthe approximated quantity and suggest that the new \napproximation in Eq.~(\\ref{eq:closure_ISI_new}) is \nsuperior to the relation in \nEq.~(\\ref{eq:closure_ISI_old}). Note that the only time \nthe homogeneity closure appears to perform better is when \nit intersects the measured value of $\\isi\/\\s$, and, \ntherefore, its superiority over the performance of the \nnew closure is rather coincidental. Further consideration \nof the results in Fig.~\\ref{fig:ISI_W} shows that, as we \nmove away from the derivation regime of slow \nrewiring rates, \nthe performance of the new closure diminishes, though it \nis still superior to the old approximation. Predictably, \nas shown in Fig.~\\ref{fig:ISI_P}, the performance of the \nnew approximation improves for the larger values \nof infection rate, and outperforming \nthe original closure even near the epidemic threshold.\n\nFinally, we test our new closure outside of the steady-state. Thus,\nFig.~\\ref{fig:sis_time} compares the performance of the newly derived\napproximation to that of the homogeneity approximation. We can see that, unlike\nthe homogeneity closure, the new closure follows the measured values of $\\isi\/\\s$\nvery accurately. Furthermore, as shown in Fig.~\\ref{fig:sis_time_triple}, the\nclosure of Eq.~(\\ref{eq:closure_ISI_new}) performs better as the solution approaches the\nsteady-state.\n\n \\begin{figure}[tb!]\n \\subfigure[]{\\includegraphics[]{Shkarfig10a}\\label{fig:sis_time}}\n \\subfigure[]{\\includegraphics[]{Shkarfig10b}\\label{fig:sis_time_triple}}\n \\caption{Time evolution of ISI closure. \nFig.~\\subref{fig:sis_time} shows the time evolution of \nthe number of ISI triples per S-node (circles, red \nonline) and the approximate value as obtained from the \nrelation in Eq.~(\\ref{eq:closure_ISI_old}) (light curve, \ngreen online) and from Eq.~(\\ref{eq:closure_ISI_new}) \n(dark curve, black online). \nFig.~\\subref{fig:sis_time_triple} shows the relative \nerror due to the two approximations. The simulations are \nperformed for $w=10^{-1}$, $p=10^0$, $r=1$, $10^5$ nodes \nand $5\\times 10^5$ links. The initial network is a \nrealization of an Erd\\\"{o}s-R\\'{e}nyi random network; the \nstate of each node is randomly assigned, with 90\\% \nI-nodes and 10\\% S-nodes. We take the average over 100 \ndynamical realizations.\n}\n\\label{fig:SIS_transient}\n \\end{figure}\n\n\n \\section{Discussion}\n\n We presented an approach for closing a mean-field description of dynamical network systems. In our approach we\nproposed exploiting the possible simplification of the heterogeneous mean-field description of the system in some\nasymptotic regime. We applied this approach to two examples of adaptive networks: recruitment to a cause model\nand a model of epidemic spread on an adaptive network. Using the two examples, we successfully developed\nclosures that perform as well as or better than the usual closures, which are based on the assumptions of\nhomogeneous distribution of nodes throughout the network.\n\n The closure we developed for the recruitment \nmodel showed significant improvement of the \nmean-field description over the one where all of \nthe high order terms were approximated using the \nhomogeneity closure. Not only do we see an \nimprovement in the predicted levels of the \nrecruited population; we also see greater \nconsistency between the moment closure \napproximation and direct measurements of the \nclosed terms. Thus, out of the three \nnode-triple terms that we approximated, one \nshowed significant improvement over the \nhomogeneity based closure, and the sum of the \nremaining two triples proved to be consistent \nwith the homogeneity closure.\n\n\nIn case of the epidemic model, the closure developed with the asymptotic approach also showed improvement over\nthe ad hoc, homogeneity based closure. The result of utilizing our approach was an improved moment closure\napproximation for one of the terms, contingent on improvements of a closure for the other term, as confirmed\nby the numerical simulations of the adaptive system.\n\nIt is important to note that the closures that we derived \nin some asymptotic regimes proved to be more accurate \nthan the homogeneity closures even outside of the \nderivation limit. For example, even though in both cases \nthe closures were derived at steady-state, they showed \nexcellent results outside of the asymptotic parameter \nregime where the derivation took place, as well as during \nthe transient state of the dynamical process. The \nadditional benefit of using this approach is that it \nallows us to expect good performance of the closure at \nleast in the limit where the derivation took place, more \nthan can be said about any ad hoc moment closure \napproximation. However, the more rigorous\nstatements about the accuracy of this approach as well as \nthe applicability of this approach to a more general \nclass of network problems are left to the future \ninvestigations.\n\n\n\n \\begin{acknowledgments}\n This work was supported by the Army Research \nOffice, Air Force Office of Scientific Research, \nby Award Number R01GM090204 from the National \nInstitute Of General Medical Sciences. MSS was also \nsupported by \nthe US National Science Foundation through grant \nDMR-1244666. The content is solely the \nresponsibility of the authors and does not \nnecessarily represent the official views of the \nNational Institute of General Medical Sciences \nor the National Institutes of Health.\n\n \n\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA stably-stratified atmosphere, characterized by a positive vertical entropy gradient, can support gravity waves, or $g$-modes.  Gravity waves are oscillations which arise from the buoyancy of parcels in the fluid. Such waves are readily excited by flow over thermal and surface topography, convective and shear instabilities, and flow adjustment processes.  They propagate through the atmosphere both horizontally and vertically.  Gravity waves in the terrestrial atmosphere and ocean are much studied \\citep[e.g.,][]{Gossard1975,Gill1982}.  They have also been observed on other Solar System bodies, such as Jupiter \\citep{Young1997} and Venus \\citep{Apt1980}.\n\nIn the terrestrial atmosphere, a typical gravity wave has an energy flux of approximately 10$^{-3}$ to 10$^{-1}$~W~m$^{-2}$.  Despite being small, compared to the total amount of absorbed solar flux ($\\sim$237~W~m$^{-2}$), gravity waves are responsible for significantly modifying---even dictating---large-scale flow and temperature structures.  Several well known examples of this are the Quasi-Biennial Oscillation, reversal of mean meridional temperature gradient in the upper middle atmosphere, and generation of turbulence \\citep[e.g.,][]{Andrews1987}.  We expect similar effects to be present in the atmospheres (and oceans) of extrasolar planets.  Moreover, due to the greater irradiation and scale heights on them, the acceleration and heating effects of gravity waves can be much stronger on hot extrasolar planets.\n\nThere has been much interest in modeling the atmospheric circulation of extrasolar planets \\citep[e.g.,][]{Joshi1997,Showman2002,Cho2003,Cho2008a,Burkert2005,Cooper2005,Dobbs-Dixon2008,Koskinen2007,Langton2007,Langton2008,Menou2009,Showman2008}.  Accurate simulation of atmospheric circulation is crucial for interpreting observations of extrasolar planets, as well as for improving theoretical understanding in general.  For this, the role of eddies and waves in transferring momentum and heat needs to be addressed \\citep{Cho2008b}. This has long been recognized in Solar System planet studies \\citep[e.g.,][]{Lindzen1990,Fritts2003}.\n\nThe plan of the paper is as follows.  In \\S\\ref{sec:theory} we derive the governing equation appropriate for linear monochromatic gravity waves on hot extrasolar planets. We also discuss a simple parameterization of the key non-linear process, saturation.  In addition, we present solutions to the equation for simple isothermal atmospheres, with and without shear in the background mean flow.  In \\S\\ref{sec:application} we extend the calculation to a physically more realistic situation, by using background flow and temperature profiles derived from a three-dimensional (3-D) hot--Jupiter atmospheric circulation simulation.  This is the first such calculation to have been performed for extrasolar planets.  Through this, the significant effects of gravity waves on hot extrasolar planet atmospheric mean flows are demonstrated.  In this section, we also discuss a way in which gravity waves can transport momentum and heat horizontally---e.g., from the dayside to nightside on tidally locked planets.  In \\S\\ref{sec:implications} we discuss the implications of our work for current extrasolar planet atmospheric modeling work.  We conclude in \\S\\ref{sec:conclusion}.\n\n\\section{Linear Theory}\\label{sec:theory}\n\n\\subsection{\\TG Equation}\\label{tge} \n\nThe dynamics of a linear gravity wave is described by the {\\it   Taylor--Goldstein equation} (TGE).  This equation is derived from the full, 3-D hydrodynamics equations \\citep{Batchelor1967}. In this work, we restrict the description to two dimensions and neglect rotation:\n\\begin{subequations}\\label{hydroeqn}\n\\begin{eqnarray}\n  \\frac{\\rD \\vec{u}}{\\rD t}\\ & = &\\ -\\frac{1}{\\rho}\\mbox{\\boldmath{$\\nabla$}} p + \n  \\vec{g}   \\\\ \n  \\frac{\\rD \\rho}{\\rD t}\\ & = &\\ -\\rho\\mbox{\\boldmath{$\\nabla$}} \\cdot \\vec{u} \\\\\n  \\frac{\\rD \\theta}{\\rD t}\\ & = &\\ \\frac{\\theta}{c_p T}\\dot{Q}\\, , \\end{eqnarray}\n\\end{subequations}\nwhere $\\rD \/ \\rD t = \\upartial \/ \\upartial t + \\vec{u} \\cdot \\mbox{\\boldmath{$\\nabla$}}$; $\\vec{u} = (u,w)$ is the flow in the horizontal and vertical directions $(x,z)$, respectively; $\\rho$ is the density; $\\vec{\\mbox{\\boldmath{$\\nabla$}}} = (\\upartial \/ \\upartial x, \\upartial \/ \\upartial z$); $p$ is the pressure; $\\vec{g} = (0,-g)$ is the gravity; $\\theta$ is the potential temperature; $c_p$ is the specific heat at constant pressure; $T$ is the sensible temperature; and $\\dot{Q}$ is the net diabatic heating rate.  Equations~(\\ref{hydroeqn}) are supplemented with the ideal gas law:\n\\begin{equation} \n  p = \\rho R T\\, ,\n\\end{equation}\nwhere $R$ is the specific gas constant.  Note that\n\\begin{equation} \n  \\theta \\equiv \n  T \\left(\\frac{p_{\\scriptscriptstyle R}}{p}\\right)^{\\kappa},\n\\end{equation}\nwhere $p_{\\scriptscriptstyle R}$ is some reference pressure (here taken to be the pressure at the lower boundary of the model) and $\\kappa = R\/c_p$; $\\theta$ is related to the entropy $s$ by $\\rd s = c_p\\,\\rd\\ln\\theta$.\n\nThe neglect of the third dimension and rotation requires that we restrict our analysis to waves with horizontal scale $L \\la U\/\\Omega$, where $U$ is the characteristic mean flow speed and $\\Omega$ is the planetary rotation rate.  This scale is adequate for all gravity waves, except for large-scale tides (which has been recently considered by \\citet{Gu2009}).  As an example, $U\/\\Omega \\sim 10^7$~m for \\HD, based on $U$ in hot extrasolar planet simulations of \\citet{Thrastarson2009}.  The resulting value is approximately 1\/10 of the planet's radius.  In addition, $\\vec{g}$, $R$, and $c_p$ are taken to be constant and $\\dot{Q}$ is specified.  These restrictions do not mitigate the basic application and implications presented in this work.  However, for broader applications, relaxation of these and other restrictions will be considered in future work.\n\nThe variables in equations~(\\ref{hydroeqn}) are all expanded as a small perturbation about a mean value, which is a function of height only: \n\\begin{equation}\\label{expan} \n  \\zeta(x,z,t)\\ =\\ \\zeta_0(z) +   \\zeta_1(x,z,t)\\, .\n\\end{equation}\nFor the thermodynamic variables, we require that $\\zeta_1 \/ \\zeta_0 \\ll~1$; however, this is not required for the flow variables, $u$ and $w$.  The mean state is assumed to be in hydrostatic balance, $\\rd p_0\/\\rd z = -\\rho_0\\, g$, and contains only horizontal flow so that $w_0 = 0$.  We also assume the anelastic approximation \\citep[e.g.,][]{Ogura1962}, $\\mbox{\\boldmath{$\\nabla$}}\\cdot(\\rho_0\\vec{u}) = 0$.  This implies the following:\n\\begin{subequations}\\label{anelas}\n  \\begin{eqnarray}\n    \\frac{N^2 H^2_{\\rho}}{c^2_s}\\ & \\ll &\\ 1 \\\\ \n    \\frac{\\gamma D}{H_p}\\ & \\la &\\ 1 \\\\\n    \\left| \\frac{u_0}{w_1} \\right| \\frac{D}{L}\\ & \\la &\\ 1, \n  \\end{eqnarray}\n\\end{subequations}\nwhere $N(z) = [g\\,(\\rd \\ln \\theta_0 \/ \\rd z)]^{1\/2}$ is the \\BV\\ frequency; $H_{\\rho}(z) \\equiv |\\rho_0\\,(\\rd \\rho_0\/\\rd z)^{-1}|$ and $H_p(z) \\equiv |p_0\\,(\\rd p_0\/\\rd z)^{-1}|$ are the density and pressure scale heights, respectively; $\\gamma =~c_p\/c_v$ is the ratio of specific heats, with $c_v$ the specific heat at constant volume; $c_s=(\\gamma RT)^{1\/2}$ is the speed of sound; and, $D$ is the vertical scale of the motion.\n\nWith (\\ref{expan}) and (\\ref{anelas}), we obtain\n\\begin{subequations}\\label{linear}\n\\begin{eqnarray} \n  \\frac{\\upartial u_1}{\\upartial t} + \n  u_0 \\frac{\\upartial u_1}{\\upartial x} + \n  w_1 \\frac{\\rd u_0}{\\rd z}\\ & = & \\ \n  -\\frac{\\upartial  \\Phi_1}{\\upartial x} \\\\\n  \\frac{\\upartial w_1}{\\upartial t} + \n  u_0 \\frac{\\upartial w_1}{\\upartial x}\\ & = &\\  \n  -\\frac{\\upartial\\Phi_1}{\\upartial z} + g \\Theta_1 \\\\ \n  \\rho_0 \\frac{\\upartial u_1}{\\upartial x} + \n  \\rho_0\\frac{\\upartial w_1}{\\upartial z} + \n  w_1\\frac{\\rd\\rho_0}{\\rd z}\\ & = &\\  0 \\\\\n  \\frac{\\upartial \\Theta_1}{\\upartial t} + \n  u_0\\frac{\\upartial \\Theta_1}{\\upartial x} + \n  w_1\\frac{\\rd \\ln\\theta_0}{\\rd z}\\ & = & \\ \n  \\frac{\\dot{Q}}{c_p T_0}\\, ,\n\\end{eqnarray}\n\\end{subequations}\nwhere $\\Phi_1 = p_1 \/ \\rho_0$ and $\\Theta_1 = \\theta_1 \/ \\theta_0$. Since the coefficients in equation~(\\ref{linear}) are independent of $x$ and $t$, we assume perturbations of the form,\n\\begin{equation}\n  \\zeta_1(x,z,t)\\ =\\ \\tilde{\\zeta}(z)\\,\\exp\\{i(kx - \\omega t)\\}\\, ,\n\\end{equation}\nwhere it is understood that the real part is to be taken.  This leads to the polarization equations:\n\\begin{subequations} \n\\begin{eqnarray}\n  -ik \\left(c-u_0 \\right) \\tilde{u} + \\frac{\\rd u_0}{\\rd z} \\tilde{w}\\   \n  & = &\\ -ik \\tilde{\\Phi}\\\\\n  -ik \\left(c-u_0 \\right) \\tilde{w}\\ & = &\\ \n  -\\frac{\\rd \\tilde{\\Phi}}{\\rd z} + g \\tilde{\\Theta} \\\\\n  -ik \\tilde{u}\\ & = &\\ \\frac{\\rd \\tilde{w}}{\\rd z} -             \\frac{\\tilde{w}}{H_{\\rho}} \\label{eq:u1polar} \\\\\n  -ik \\left(c-u_0 \\right) \\tilde{\\Theta} + \\frac{N^2}{g} \\tilde{w}\\ \n  & = &\\ \\tilde{F},\n\\end{eqnarray}\n\\end{subequations}\nwhere $F \\equiv \\dot{Q}\/(c_pT_0)$ is the forcing, $c = \\omega\/k$ is the constant (possibly complex, see \\S\\ref{sec:saturation}) horizontal phase speed, and $(c - u_0)$ is the intrinsic phase speed.  Now, transforming to a new variable so that the effect of decreasing density with height is compensated,\n\\begin{subequations}\\label{eq:newvar}\n\\begin{eqnarray} \n  \\hat{w}(z)\\ & = &\\ \\tilde{w}\\, \\exp\\{ -\\chi(z) \\} \\\\ \n  \\chi(z)\\ & = &\\ \\int^z_{z_b} \\frac{\\rd \\xi}{2H_{\\rho}(\\xi)}\\, ,   \n\\end{eqnarray} \n\\end{subequations}\nwhere $z_{\\rm b}$ is $z$ at the bottom, we obtain the TGE:\n\\begin{equation}\\label{eq:TGE}  \n  \\frac{\\rd^2 \\hat{w}}{\\rd z^2} + m^2 \\hat{w}\\ =\\     \\frac{\\kappa\\,\\dot{Q}}{H_p \\left(c-u_0\\right)^2}\\, e^{-\\chi}\\, .\n\\end{equation}\nHere, $m = m(z)$ is the index of refraction, and corresponds to the local vertical wavenumber. It is given by\n\\begin{equation}\\label{eq:indref} \n  m(z)\\ =\\ \\left[\\frac{N^2}{\\left(c-u_0\\right)^2} +                       \\frac{u_0''}{\\left(c-u_0\\right)} +                           \\frac{u_0'}{H_{\\rho}\\left(c-u_0\\right)} - \\frac{1}{4H^2_{\\rho}} -         k^2 \\right]^{1\/2}. \n\\end{equation} \nIn equation~(\\ref{eq:indref}) we have used $H_p = RT_0(z)\/g$ and the prime indicates differentiation with respect to $z$. \n\nThe vertical structure of the perturbations, oscillating in $z$, signify that we are dealing with {\\it internal} waves.  In equation~(\\ref{eq:indref}), the key terms contributing to $m^2$ are the first (``buoyancy'') term and the last (``non-hydrostatic'') term. Although the other three terms contribute, generally the buoyancy and non-hydrostatic terms control whether the wave propagates since the flow shear and curvature are small and the scale height is large in practice.  For large wavelength waves, the waves are hydrostatic and the non-hydrostatic term is small; then, the buoyancy term dominates. In these cases, as long as the atmosphere is stratified (i.e., $N^2 > 0$) and $c \\neq u_0$, the wave will propagate vertically. However, for shorter, non-hydrostatic waves, it is possible that $k^2 > [N \/ (c - u_0)]^2$. In these cases, $m$ is imaginary and the wave does not propagate. This situation is discussed in more detail in \\S\\ref{subsec:WBI}.\n\nEquation~(\\ref{eq:TGE}) can be thought of as a driven harmonic oscillator equation.  When $m$ is real and constant, its solution is a simple sinusoid. Since the transformation, equation~(\\ref{eq:newvar}), compensates for the exponential decay of density with height, $w$ grows (decays) rapidly with height (depth).  Here, we have dropped the tilde overscript.  From hereon we drop all tilde overscripts and ``1'' subscripts for notational clarity.  The growth can be clearly seen in Figure~\\ref{fig:noshear}, along with the corresponding constant stress (vertical transport of horizontal momentum) when there is no dissipation (see~\\S\\ref{sec:saturation}).\n\n\\begin{figure}\n  \\epsscale{1.08}\n  \\plotone{Paper1Fig0.eps}\n \n  \\caption{A gravity wave propagating in an isothermal ($T_0 = 1350$~K) and     constant background flow ($u_0 = 350$~m~s$^{-1}$) atmosphere.  The phase     speed of the wave $c$ is 100~m~s$^{-1}$ and the horizontal wavelength,     $2\\pi \/ k$, is 2500~km.  The vertical perturbation velocity $w$     ($\\cdots$), horizontal perturbation velocity $u$~(---), and wave stress     $\\tau$ ($-\\! \\cdot\\!  -$) are shown.  The latter is the vertical transport     of the horizontal momentum.  Wave amplitudes, $u$ and $w$, grow     exponentially with height, but the wave stress is constant with height,     since there is no dissipation.  The jump in $\\tau$ near $z\/H_p = 1$ is     caused by the forcing. \\label{fig:noshear}}\n\\end{figure}\n\nIf the temperature or the flow varies with height, $m$ is a function of height $z$.  If $m$ varies slowly, the WKB approximation \\citep{Bender1999} can then be used to obtain\n\\begin{equation}\\label{eq:WKBsoln} \n  w(z)\\ \\approx\\ \n  \\frac{A\\, e^{\\, z\/2H_{\\rho}}}{m^{1\/2}}                             \\exp\\left\\{\\pm\\, i\\! \\int_{z_{\\rm b}}^z m(\\xi)\\, \\rd \\xi\\right\\}\\, . \n\\end{equation}\nHere, $A = w(z_{\\rm b})\\,[m(z_{\\rm b})]^{1\/2}$.  As in the constant $m$ case, the vertical perturbation velocity is wave-like with upwardly and downwardly propagating components; the amplitude of the upward component grows with height and the downward component decays with depth.  However, when $m$ varies rapidly, the solution must be obtained numerically.\n\nAt the boundaries we use the radiation condition, selecting the upwardly propagating solution at the top boundary $z_t$ and the downwardly propagating solution at the lower boundary $z_b$.  This is achieved using the condition,\n\\begin{equation}\\label{eq:BC}\n  \\frac{\\rd \\hat{w}}{\\rd z} - \\left(ism + \\dfrac{1}{2m} \\dfrac{\\rd m}{\\rd z} \\right) \\hat{w} \\ =\\ 0.\n\\end{equation}\nHere, $s=\\pm 1$, depending on the signs of the horizontal and intrinsic phase speeds and on whether the condition is at the upper or lower boundary.  Note that condition~(\\ref{eq:BC}) requires the WKB approximation to be valid at the boundaries.  For example, the boundaries cannot be critical layers, regions where $(c - u_0)\\rightarrow 0$, since $m \\rightarrow\\infty$ approaching such a layer and the WKB approximation ceases to be valid.  Critical layers will be discussed further below.\n\nAs alluded to above, we use Gaussian elimination \\citep{Canuto2006} in this work to numerically solve the TGE.  In the cases presented here, 3000 equally spaced levels are employed to solve for $w$.  We have checked that this resolution is adequate by performing calculations with 10,\\! 000 levels and verifying convergence at the higher resolution. Extensive validations against known analytic solutions, where they exist, have also been performed.\n\n\\subsection{Polarization Relations and Fluxes}\\label{subsec:pol}\n\nThe solution to the TGE is a wave in the vertical velocity perturbation.  This can be related to the horizontal and temperature (sensible and potential) perturbations.  Understanding these are essential for parameterizing the saturation process in the full non-linear situation and in general circulation models.  The perturbation quantities can be obtained from the polarization equations \\citep{Hines1960}:\n\\begin{subequations}\\label{eq:polar}\n  \\begin{eqnarray} \n    u\\ & = &\\ \\frac{i}{k} \\left(\\frac{\\rd w}{\\rd z} -                   \\frac{w}{H_{\\rho}}   \\right) \\\\\n    \\Phi\\ & = &\\ \\frac{i}{k}\\left[ \\left( c - u_0 \\right)                   \\left(\\frac{\\rd w}{\\rd z} -\\frac{w}{H_{\\rho}} \\right) +\n      \\frac{\\rd u_0}{\\rd z} w \\right] \\\\\n    \\theta\\ & = &\\ -\\frac{i}{k}\\left[\\frac{\\theta_0 N^2}\n      {g \\left(c - u_0\\right)}\\right] w \\\\\n    T\\ & = &\\ -\\frac{i}{k}\\left[\\frac{T_0 N^2} {g \\left(c -           u_0\\right)}\\right] w\\, ,\n\\end{eqnarray}\n\\end{subequations}\nwhere we have introduced the geopotential perturbation function $\\Phi$ ($\\equiv p\/\\rho_0$).  Equation~(\\ref{eq:polar}a) gives the relationship between the vertical and horizontal velocity perturbations.  Note that the amplitude of $u$ is larger than the amplitude of $w$, as it is scaled by $kH_\\rho$.  Note also that the geopotential (pressure) perturbation varies with the background flow via the dependence on the intrinsic phase speed, according to equation~(\\ref{eq:polar}b).  Equations~(\\ref{eq:polar}c) and (\\ref{eq:polar}d) shows that the potential and sensible temperature perturbations, respectively, are both $\\pi\/2$ out of phase with the vertical velocity perturbation.  But, the phase between $u$ and $w$ varies locally through their dependence on the background.\n\nGravity waves are an efficient means of transporting both momentum and energy.  The (perturbation) momentum and energy fluxes are simply obtained from equations~(\\ref{eq:polar}):\n\\begin{subequations}\\label{eq:flux}\n\\begin{eqnarray}\n  \\tau\\ & = &\\ \\rho_0\\,\\overline{uw} \\\\\n  F_x\\ & = &\\ \\rho_0\\,\\overline{\\Phi u} \\\\\n  F_z\\ & = &\\ \\rho_0\\,\\overline{\\Phi w} \\, .\n\\end{eqnarray}\n\\end{subequations}\nIn equations~(\\ref{eq:flux}), $\\tau$ is the vertical flux of horizontal momentum (or, the wave stress); $F_x$ and $F_z$ are the horizontal and vertical fluxes of energy, respectively; and, the overline indicates an average over a wavelength,\n\\begin{equation} \n  \\overline{\\alpha \\beta}\\ =\\ \n  \\frac{1}{2}\\,{\\rm Re}\\left\\{ \\alpha \\beta^* \\right\\}\\, ,\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are arbitrary complex functions and ``*'' denotes the complex conjugate.  Note that the energy fluxes depend on the background flow through their dependence on $\\rho_0$ and $\\Phi$.  However, as can be seen in Figures~\\ref{fig:noshear} and \\ref{fig:excritlayer}, the wave stress remains constant, away from the forcing and damping regions---e.g., saturation regions and critical layers.  This is in accordance with the second Eliassen--Palm theorem \\citep{Eliassen1960}, which expresses non-interaction of the disturbance in the absence of dissipation and forcing.\n \nIn Figure~\\ref{fig:excritlayer}, it is important to note that, where the fluxes are changing, the wave is interacting with the background flow in those regions.  This should be contrasted with the behavior illustrated in Figure~\\ref{fig:noshear}, where the stress is not changing.  Changes in the wave stress cause accelerations to the mean flow.  Correspondingly, changes in the energy fluxes cause the temperature of the region to change.  The rates of these changes are given by \\begin{subequations}\\label{eq:rate} \\begin{eqnarray}\n  \\frac{\\upartial u_0}{\\upartial t}\\ & = & \\ -\\frac{1}{\\rho_0} \\frac{\\upartial \\tau}{\\upartial z} \\\\\n  \\frac{\\upartial T_0}{\\upartial t}\\ & = & \\ -\\frac{1}{\\rho_0 c_p} \\frac{\\upartial F_z}{\\partial z}\\, . \n\\end{eqnarray}\n\\end{subequations}\nIn this case, the wave causes the background flow to accelerate from 350~m~s$^{-1}$ towards 500~m~s$^{-1}$.\n\nIn the remainder of the paper, when speaking of vertically propagating waves, we consider only waves that propagate upwards from the region of excitation.  However, it must be remembered that downward propagating waves are also generated.  On a giant planet without a solid surface, those waves may not be reflected or absorbed.  They can continue to penetrate downward until they encounter a critical level or a convective region.  Or, they are simply dissipated since the amplitudes of the downwardly propagating waves decrease exponentially, as already discussed (and as also can be seen in Figure~\\ref{fig:excritlayer}).  The energy and momentum fluxes are linked by the first Eliassen--Palm theorem \\citep{Eliassen1960}:\n\\begin{equation} \\label{eq:ep1}\nF_z = \\left(c - u_0 \\right) \\tau,\n\\end{equation}\nwhich can be derived from equations~(\\ref{eq:polar}) and (\\ref{eq:flux}).  For downwardly propagating waves, we have $F_z < 0$; and, therefore, via equations~(\\ref{eq:ep1})~and~(\\ref{eq:rate}a), we see that for these waves the deposition of momentum also leads to acceleration of the flow toward the phase speed of the wave.  Downwardly propagating planetary scale gravity waves (i.e., thermally excited tides) are considered by \\citet{Gu2009}.\n\n\\subsection{Saturation and Critical Layers}\\label{sec:saturation}\n\n\\begin{figure}\n  \\epsscale{1.1} \n  \\plotone{Paper1Fig2.eps}\n  \\caption{As in Figure~\\ref{fig:noshear}, but with negative \n   vertical wind shear: $u_0$ varies from 350~m~s$^{-1}$ near \n  the bottom to 650~m~s$^{-1}$ near the top, with linear growth in between.  Here, $c = 500$~m~s$^{-1}$; hence, $(c - u_0) \\rightarrow 0$ at $z\/H_p \\approx 8$, where the wave encounters a critical layer and dissipates.    In the region just below the critical layer, the     wave saturates and transmits momentum into the background flow, as indicated by the drop in $\\tau$ with height there.       \\label{fig:excritlayer}}\n\\end{figure}\n\nThe present work concerns inviscid, linear, monochromatic waves.  Such waves are infinite in extent and, in principle, can grow without limit when they propagate upward.  This is obviously not physical.  In reality, such waves become unstable and saturate.  The saturation process can be treated by introducing a correction to the solution according to the physical criterion,\n\\begin{equation} \\label{eq:satcrit1} \n\\frac{\\upartial }{\\upartial z} \\left(\\theta_0 + \\theta \\right)\\ \\leq \\ 0,\n\\end{equation}\nensuring that a wave does not become convectively unstable and remains at neutral stability.  When the saturation region is identified, $\\theta$ is adjusted so that neutral stability is maintained.  This new value for $\\theta$ is then used in equation~(\\ref{eq:polar}c) to obtain the corresponding $w$ in the saturated wave.  This value of $w$ is then used in equations~(\\ref{eq:polar}a, b, c), (\\ref{eq:flux}) and (\\ref{eq:rate}).  This saturation condition acquires the simple form,\n\\begin{equation} \\label{eq:satcrit2}\n  \\left| u \\right| \\ = \\ \\left| c - u_0 \\right|,\n\\end{equation}\nin situations where the WKB approximation is valid \\citep{Fritts1984}.\n\nAs can be seen from the definitions of $N$, $H_p$, and $\\theta$, the \\BV\\ frequency can be written as\n\\begin{equation} \\label{eq:BVF} \nN \\left(z\\right) = \\left[ \\frac{g}{T} \\left(\\frac{\\upartial T}{\\upartial z} + \\frac{g}{c_p} \\right)   \\right]^{1\/2}.\n\\end{equation}\nSince $g$ and $c_p$ are essentially constants in the modeled height range, $N$ depends only on the temperature profile $T(z)$.  For isothermal regions, $N$ is a constant.  In general, the fractional change of $T$ with height is small compared to $g\/(Tc_p)$ throughout the modeled region.  Hence, $N$ is nearly constant in the entire domain, with a value that is roughly $2.4\\!\\times\\! 10^{-3}$~s$^{-1}$.  The \\BV \\ frequency for our atmospheric profile is shown in Figure~\\ref{fig:flowtempprof}.  Further discussion of the effects of variation in $N$ are presented in \\S\\ref{subsubsec:background}.\n\nIf the background flow contains shear, it is possible for the wave to encounter a critical layer at some height.  At the critical level, where $c = u_0$, the TGE becomes singular.  However, the equation can be solved using the method of Frobenius \\citep{Bender1999}, from which it is seen that the wave is drastically attenuated by the critical layer \\citep{Booker1967}.  The amount of attenuation depends on the Richardson number $Ri$ of the flow,\n\\begin{equation} \n  Ri\\ =\\ \\frac{N^2}{\\left(\\rd u_0\/\\rd z\\right)^2}\\ . \n\\end{equation}\n\nFigure~\\ref{fig:excritlayer} illustrates a critical layer encountered by a wave. If the wave stress has magnitude $\\tau$ below the critical layer, it then has magnitude, $\\tau \\cdot \\exp \\left\\{-2 \\pi \\left[Ri - (1\/4)\\right]^{1\/2} \\right\\}$, after the encounter with the critical layer \\citep{Booker1967}.  This is a substantial amount of attenuation.  For example, in our model atmosphere of Figure~\\ref{fig:excritlayer}, $Ri \\ga 900$.  Hence, the wave is essentially completely dissipated at the critical level, with the wave stress falling to practically zero and the momentum deposited into the mean flow.  Note that, during its approach to the critical layer, the wave actually saturates and deposits momentum and energy over a finite layer (cf., Figure~\\ref{fig:excritlayer}).  It is important to note that while a wave saturates when approaching a critical layer, the presence of a critical layer is {\\it not} required for saturation.  Saturation is a general process referring to wave dissipation by many different mechanisms---e.g., radiation, conduction, breaking, and turbulence.\n\nNumerically, when a critical layer is present, we lift the phase speed from the real axis by adding a small imaginary component: $c = c_r + ic_i$, where $|c_i \/ c_r| < 10^{-3}$.  This introduces a small amount of linear damping and ensures that the neglected nonlinear terms do not dominate in the regions where waves become steep and eventually break.  Of course, adding damping causes the wave-stress to decrease with height and the second Eliassen-Palm theorem to be no longer valid.  However, the effect is small.  This can be seen in Figure~\\ref{fig:excritlayer}, where the wave stress falls negligibly over the layer from $z\/H_p \\approx 1$ to $z\/H_p \\approx 5$.\n\n\\section{Extrasolar Planet Application}\\label{sec:application}\n\n\\subsection{Setup}\n\nOur aim in the present paper is to demonstrate several properties of gravity waves likely to be important for hot extrasolar planets.  For this, we choose \\HD\\ as a paradigm planet. This planet has been the focus of many theoretical and observational studies, and it is expected to be generic with respect to the properties discussed here.  The physical parameters that characterize the planet's atmosphere are given in Table \\ref{tab:params}.\n\n\\begin{table}\n\\begin{center}\n  \\caption{Parameters Used for ${\\rm HD}\\,209458\\,{\\rm b}$}\n\\begin{tabular}{llr}\n  \\hline\n  \\\\\n  Specific Gas Constant & $R$ & 3523~J~kg$^{-1}$~K$^{-1}$ \\\\\n  Specific Heat at Constant Pressure \\hspace*{5mm} & $c_p$             \\hspace*{3mm} &   12300~J~kg$^{-1}$~K$^{-1}$ \\\\\n  Acceleration Due to Gravity & $g$ & 10~m~s$^{-2}$ \\\\\n  Rotation Rate & $\\Omega$ & 2.08$\\times 10^{-5}$~s$^{-1}$ \\\\\n  \\\\\n  \\hline\n\\end{tabular}\\label{tab:params}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Forcing}\n\nIn a stratified atmosphere, gravity waves are readily generated by many mechanisms---thermal and mechanical---such as the absorption of stellar radiation, convective release of latent heat, storms, flows over topography and coherent localized ``heat islands'', and detonation by impacts. In this work, we consider small- and meso-scale thermally excited waves, rather than large-scale waves, as already discussed.  The horizontal wavelengths used in this work are 2500~km or less.  This is a reasonable range since it is well within the observed range of gravity waves on Jupiter \\citep{Young1997,Hickey2000}.  Although not unimportant, we do not dwell on the precise nature of the source of the excitations. The main interest here is in the propagation and deposition of momentum and energy.\n\nThe forcing in equation~(\\ref{eq:TGE}) is simply represented as a Gaussian, modified so that it is zero beyond two half-widths above and below the center. The center is located at $z\/H_p = 1$ above the bottom of the domain.  The half-width is 75~km, or $\\sim0.15 H_p$.  The forcing location and width are chosen so that the vertical scale of the forcing is less than the vertical wavelength of the waves we present here.  We have extensively explored the effects of varying the location, width, and strength of forcing and present this case to illustrate several important points.  Not surprisingly, the dynamics do depend on the chosen parameter values, but the dependence is broadly predictable.  For example, if the forcing scale is much larger than the vertical wavelength, then only a very small amplitude wave is emitted from the forcing region, due to cancellations.\n\nThe heating rate, $\\dot{Q} \/ c_p$, is set to 10$^{-3}$~K~s$^{-1}$. Note that this is a modest value.  The forcing corresponds to roughly 300~K per rotation of the planet.  This is compared to $\\sim$100~K per rotation at the chosen location in the circulation model of \\citet{Thrastarson2009}.  A forcing of $\\sim$1100~K per rotation, for a similar latitude-longitude location on the planet (see \\S\\ref{subsubsec:background}), is used in \\citet{Showman2008}. The latter value implies that, in the absence of motion, the location on the planet will cool completely in one rotation of the planet.  We stress that locally---i.e., scales far below the grid scale of the current circulation models---the forcing could actually be much stronger. The actual value is presently uncertain and likely to be spatially and temporally variable over the planet. To provide a context, for the Earth the heating rate is $\\sim\\! 2$~K per day (1 day = 0.29 rotation of \\HD) over large areas; but, locally, at the tops of low clouds on the Earth the rate can be up to $\\sim\\! 50$~K per day \\citep{Wallace2006}.\n\n\\subsubsection{Background Structure}\\label{subsubsec:background}\n\n\\begin{figure}\n  \\vspace{2pt}\n  \\epsscale{1.1}\n  \\plotone{Paper1Fig3.eps}\n  \\caption{Sample atmospheric mean flow $u_0$ (---), temperature $T_0$     ($-\\!  \\cdot\\!  -$), and \\BV \\ frequency, $N$ ($\\cdots$) profiles     of a typical hot extrasolar planet \\HD, used in this work.  The     profile is representative of a region at approximately     70$^{\\circ}$E,\\,10$^{\\circ}$N.  The     planet is a close-in giant planet.  The profiles are obtained from     a 3-D, global circulation model, up to $\\sim\\! 10^{-3}$~bar level.     Above that level the profiles are simply extended, following     loosely the observed profiles of Jupiter     \\citep{Young2005,Flasar2004}.  \\label{fig:flowtempprof}}\n\\end{figure}\n\nFigure~\\ref{fig:flowtempprof} shows the mean flow and temperature profiles used to obtain much of the results presented in this section. The lower part of both profiles---approximately the lower six scale heights---is taken from global circulation simulations of the hot extrasolar giant planet \\HD\\ by \\citet{Thrastarson2009}, using the NCAR Community Atmosphere Model \\citep{Collins2004}.  The profile is from a point near the equator, slightly away from the substellar point (70$^{\\circ}$E,\\,10$^{\\circ}$N). This point was chosen as it is within the equatorial jet and the Coriolis parameter $f= 2 \\Omega \\sin \\phi$ is not large. The temperature profile generally increases with height over the lowest 4 scale heights and then becomes isothermal. Fortuitously, this provides an opportunity for a loose validation of our model: it is very similar to the temperature structure observed by the Galileo probe in the same region of Jupiter's atmosphere \\citep{Young2005}.  We extend the profile by keeping the temperature isothermal through the planet's stratosphere and having the thermosphere (beginning of the temperature inversion near the top) start between 12 and 13 scale heights at $p_0 \\approx 4 \\times 10^{-6}$~bar.  This profile has a Richardson number of at least 3.4, giving an attenuation of $1.4 \\times 10^{-5}$.  Hence, any critical layer can be considered to fully dissipate the wave.\n\n\\begin{figure}\n  \\epsscale{1.08} \n  \\plotone{Paper1Fig4.eps}  \n  \\caption{A gravity wave with $c = 600$~m~s$^{-1}$, propagating in an     atmosphere with profiles shown in Figure~\\ref{fig:flowtempprof}.  The     horizontal perturbation velocity $u$ (---), intrinsic phase speed, $c -     u_0$ ($\\cdots$), and the mean flow acceleration $\\rd u_0\/\\rd t$ ($-\\!     \\cdot\\! -$) are shown.  The intrinsic phase speed becomes zero at $z\/H_p     \\approx 5$ and the wave encounters a critical level.  In the layers just     below the critical level the wave saturates and sheds momentum into the     mean flow, causing it to accelerate, peaking at a rate over     250~m~s$^{-1}$~rotation$^{-1}$. \\label{fig:realcritlayer}}\n\\end{figure}\n\nThe chosen flow profile has two local flow velocity maxima. The upper maximum is extended into a jet with a peak at $z\/H_p \\approx 6$. This is similar to the structure of the jet in Jupiter's stratosphere proposed by \\citet{Flasar2004}, which has a peak velocity between the 10$^{-2}$ and 10$^{-3}$~bar levels.  We then extend the profile further upwards without shear.  Note, this is in keeping with the lower boundary of the model of \\citet{Koskinen2007}.  The structure is somewhat different than those of \\citet{Showman2008,Showman2009}, where there is just one jet with the peak approximately located at the 10$^{-1}$~bar level.  The peak flow speed is also much greater in those studies at 4 or 5~km~s$^{-1}$.  It is important to note, however, that these differences do not change qualitatively the basic points we are making in this paper.\n\nThe \\BV\\ frequency profile $N(z)$ is also shown in Figure~\\ref{fig:flowtempprof}.  As already discussed, the profile does not vary much over the whole domain: the maximum \\BV\\ frequency is just 1.2 times the minimum value.  Therefore, $N$ does not contribute much to the variation of the index of diffraction $m$.  The main contributor to the variation of $m$ is the variation of the intrinsic phase speed, which is derived from the large variation of flow speeds.  This should be compared to the analogous terrestrial situation, where the range of flow speeds is much lower.  This allows $N$ to have a larger effect on the variation of $m$ on the Earth.\n\n\\subsection{Wave-Background Interaction}\\label{subsec:WBI}\n\n\\subsubsection{Critical Layer Encounter}\n\nFigure~\\ref{fig:realcritlayer} shows a gravity wave encountering a critical level in the upper jet in Figure~\\ref{fig:flowtempprof}.  The wave has a horizontal wavelength, $2\\pi\/k$, of 2500~km and $c = 600$~m~s$^{-1}$. Here, since $c > u_0$ as the wave approaches the critical layer from below, the momentum deposited in the mean flow causes the mean flow to accelerate.  This acceleration peaks at over 250~m~s$^{-1}$ per rotation.  This is large enough to double the flow speed at this layer in $\\sim$2~rotations---a significant effect.  The effect is large enough to require its inclusion in any simulation of the atmospheric circulation \\citep{Cho2008a}.\n\nThe waves encountering critical layers are dissipated.  Therefore, a flow with a range of flow speeds dissipates all gravity waves with phase speeds within this range.  That is, a spectrum of gravity waves is prevented from propagating high into the atmosphere.  There are other, secondary effects at the critical layer that may also affect the mean flow; but, they are not modeled here.  They will form the basis of future work.  For example, the deposition of energy into the flow at the critical layer may well lead to the generation of new gravity waves, which then may propagate further, partly mitigating the filtering effect.\n\n\\begin{figure}\n  \\epsscale{1.1} \n  \\plotone{Paper1Fig5.eps}\n  \\caption{A gravity wave, with $c = -40$~m~s$^{-1}$, propagating in     an atmosphere described by the profiles in     Figure~\\ref{fig:flowtempprof}.  The perturbation to the     temperature field $T$ (---) and the heating rate $\\rd T_0 \/ \\rd t$     ($-\\,\\cdot\\, -$) are shown.  The wave saturates at just above     $z\/H_p = $~10, where the heating peaks at nearly 80~K~rotation$^{-1}$. The peak energy flux for this wave is approximately 1~W~m$^{-2}$.} \\label{fig:realsaturate}\n\\end{figure}\n\n\\begin{figure}\n  \\vspace{2.5pt}\n  \\epsscale{1.1}\n  \\plotone{Paper1Fig6.eps}\n  \\caption{As in Figure~\\ref{fig:flowtempprof}, but with the bottom of     the computational domain extended down to 100~bars.  Here, $u_0$     is extended downward barotropically (independent of height) from     the 1~bar level; $T_0$ is extended downward so that the profile     below the 1~bar level is similar to that of Figure~18 in     \\citet{Showman2009}.  \\label{fig:flowtempprofdeep}}\n\\end{figure}\n\n\\subsubsection{Saturation}\n\nFigure~\\ref{fig:realsaturate} shows an example of a gravity wave saturating in the atmosphere of Figure~\\ref{fig:flowtempprof}.  In general, it is possible that a wave may not encounter a critical level as it propagates upward.  However, such a wave then travels higher into the atmosphere, where it grows large and, if unabated, eventually breaks or suffers dissipation at higher altitudes.  Although both momentum and energy are deposited in this case, we focus here on the effects on the temperature field.\n\nThe wave launched in Figure~\\ref{fig:realsaturate} has $c=-40$~m~s$^{-1}$ (i.e., westward).  The horizontal wavelength remains at 2500~km, as in the critical layer example of Figure~\\ref{fig:realcritlayer}.  The vertical velocity perturbation grows with height.  Therefore, so do the zonal velocity perturbation $u$ and the potential temperature perturbation $\\theta$, as expected from equation~(\\ref{eq:polar}).  In this case, the wave saturates near the top of the jet, where condition~(\\ref{eq:satcrit1}) is exactly satisfied.  Here, condition (\\ref{eq:satcrit2}) is approximately satisfied since the zonal perturbation velocity and the intrinsic phase speed ($c - u_0$) are both approximately 122~m~s$^{-1}$.  The saturation deposits energy that causes the atmosphere there to heat up.  The heating is significant, peaking at $\\sim\\! 75$~K per rotation---a 5\\% change in one rotation.  In the absence of other effects, the ambient temperature can be doubled in about 20 planetary rotations (or orbits, assuming 1:1 spin-orbit synchronization).\n\nA wave that has a phase speed greater than the maximum flow speed will not encounter a critical layer.  Those with phase speeds close to, but still above, the maximum flow speed will, in general, saturate in the regions just below the maximum flow, as the intrinsic flow speed will be small in that region. Similarly, waves with phase speeds just less than the minimum flow will saturate as well.  In this way the filtering effect discussed above is extended beyond those waves with phase speeds equal to flow speeds.  The main effect of these filtered waves on the flow will be lower in the atmosphere. In the profile given in Figure \\ref{fig:flowtempprof}, where the waves emanate from the $z\/H_p = 1$ level, this means that the upper layers of the lower jet will be slowed by gravity waves whereas the lower levels of the upper jet will be accelerated.\n\nThose waves that do not dissipate will be able to propagate into the upper atmosphere depositing their momentum and heat there. Here the changes to the flow can be very large.  For example, the wave shown in Figure~\\ref{fig:realsaturate} causes a deceleration of up to 6.8~km~s$^{-1}$~rotation$^{-1}$ as it saturates.  This clearly dominates the flow at this level.\n\nMoving the location of the wave origin down does not change the basic behavior in the qualitative sense.  However, the amplitudes are much larger, compared with the case when the wave originates higher up in altitude.  Thus, the possibility exists for stronger effects due to gravity wave interaction with the background.\n\nThis is illustrated in Figures~\\ref{fig:flowtempprofdeep} and \\ref{fig:realsaturatedeep}, which should be compared with Figures~\\ref{fig:flowtempprof} and \\ref{fig:realsaturate}. Here, we have extended the profiles downwards.  The wave is still launched at $z\/H_p = 1$, but this is now deeper in the atmosphere.  The wave again has a phase speed of $-40$~m~s$^{-1}$, and the horizontal wavelength remains at 2500~km.  This wave also saturates near the top of the upper jet, where the energy deposition into the mean flow causes the atmosphere there to heat up.  Note that the region of heating is lower than when the wave originates higher up, as in Figure \\ref{fig:realsaturatedeep}.  The heating is significant, peaking at $\\sim\\! 3000$~K~rotation$^{-1}$.  The ambient temperature can be doubled in approximately half of a planetary rotation.  In a more realistic scenario, dissipation---which we have not included in our model---will likely reduce the heating rate.\n \n\\begin{figure}\n  \\vspace{3pt}\n  \\epsscale{1.1}\n  \\plotone{Paper1Fig7.eps}\n  \\caption{A gravity wave, with $c = -40$~m~s$^{-1}$, propagating in     an atmosphere described by the profiles in     Figure~\\ref{fig:flowtempprofdeep}.  The perturbation to the     temperature field $T$ (---) and the heating rate $\\rd T_0 \/ \\rd t$     ($-\\,\\cdot\\, -$) are shown.  The wave saturates at just above     $z\/H_p = 13$, where the heating rate peaks at just under     3000~K per rotation.  In terms of the pressure level, this     location is actually lower than in the case shown in     Figure~\\ref{fig:realsaturate}, and the magnitude of the peak is     nearly 50~times greater. The peak energy flux for this wave is nearly 200~W~m$^{-1}$. Thus, having a source at a lower height     can have a much stronger effect.  \\label{fig:realsaturatedeep}}\n\\end{figure} \n \n\\begin{figure}\n  \\epsscale{1.08}\n  \\plotone{Paper1Fig8.eps}\n  \\caption{A gravity wave, with $c = 10$~m~s$^{-1}$ and horizontal     wavelength $2\\pi \/ k = 955$~km, trapped in an atmosphere with the     structure presented in Figure~\\ref{fig:flowtempprof}.  The     vertical velocity perturbation $w$ (---), mean flow $u_0$ ($-\\,     -$), and the real part of the index of refraction $m$ ($-\\cdot -$)     are shown.  The wave is trapped in relatively quiescent region between $z\/H_p     \\approx 1.5$ and $z\/H_p \\approx 3.5$ and does not propagate     vertically.  The region of trapping corresponds to the region     where $m$ is real.  The wave is reflected at the boundaries of     this region, providing a possibility for     resonance. \\label{fig:realtrapped}}\n\\end{figure} \n \n\\subsubsection{Refraction}\\label{subsubsec:refract}\n\nSo far we have been focusing on the vertical transport of momentum and energy by gravity waves.  However, the waves can also transport momentum and energy {\\it horizontally} (cf., \\S\\ref{subsubsec:ducting}).  Substituting $c = \\omega \/ k$ into the the index of refraction, equation~(\\ref{eq:indref}), and rearranging gives the dispersion relation for gravity waves.  Here, we consider the case where $u_0 = 0$.  We will examine cases where $u_0 \\ne 0$ in the sections following this one.\n\nWhen there is no background mean flow, the dispersion relation simplifies to\n\\begin{equation} \\label{eq:dispersion} \n\\omega\\ =\\ \\pm\\frac{Nk}{\\left[k^2 + m^2       + 1\/(4H_{\\rho}^2)\\right]^{1\/2}}.\n\\end{equation}\nWe can then use the definitions,\n\\begin{subequations} \\label{eq:grpvel}\n\\begin{eqnarray} \n  u_g &\\ =\\ & \\frac{\\upartial \\omega}{\\upartial k} \\\\\n  w_g &\\ =\\ & \\frac{\\upartial \\omega} {\\upartial m},\n\\end{eqnarray}\n\\end{subequations}\nto obtain the group velocities.  They are:\n\\begin{subequations} \\label{eq:noflowgrpvel}\n\\begin{eqnarray} \n  u_g &\\ =\\ & \\pm\\frac{N\\left[m^2 + 1\/(4H^2_{\\rho})\\right]}{\\left[k^2 + m^2 + 1\/(4H^2_{\\rho})\\right]^{3\/2}} \\\\\n  w_g &\\ =\\ & \\pm\\frac{N k m}{\\left[k^2 + m^2 + 1\/(4H^2_{\\rho})\\right]^{3\/2}}.\n\\end{eqnarray}\n\\end{subequations}\nThus, for propagating waves (i.e., waves for which $m$ is real), $u_g \\neq 0$. Therefore, gravity waves always propagate obliquely and cannot strictly propagate vertically when there is no background flow.\n\nFrom equations~(\\ref{eq:noflowgrpvel}) we see that $\\vartheta$, the angle of propagation with respect to the horizontal, is given by\n\\begin{equation}\n\\tan \\vartheta\\ =\\ \\frac{k m}{m^2 + 1\/(4H^2_{\\rho})}.\n\\end{equation}\nSince $H_{\\rho}$ is nearly constant, with a value just under 500~km, $\\tan \\vartheta$ varies with $1\/m$ for $m\\gtrsim10^{-6}$~m$^{-1}$.  This gives rise to refraction.  As a wave propagates into a region of higher $m$, it bends to a more horizontal path.  Note that, as we are here considering a region with no flow, increasing $m$ is essentially equivalent to increasing $N$.  As shown in Figure~\\ref{fig:flowtempprof}, $N$ increases in the thermosphere and so the paths of waves in this region will bend towards the horizontal even though the flow is small.\n\n\\subsubsection{Trapping}\\label{trapping}\n\nFrom equation~(\\ref{eq:WKBsoln}) we can see that in regions where $m$ is imaginary, the wave is evanescent: its amplitude decays towards zero and therefore it does not propagate vertically.  This can occur when $N^2 < 0 $ (i.e., when the atmosphere is convectively unstable).  But, it can also occur for non-hydrostatic waves when the buoyancy term becomes dominated by the non-hydrostatic term.  In addition, when the index of refraction changes between layers, the wave is reflected at the boundary.  The amount of reflection is given by the magnitude of the coefficient of reflection $|r|$, where \n\\begin{equation}\\label{eq:coeff_reflect}\n  r\\ =\\ \\frac{m_1 - m_2}{m_1 + m_2}\\, .\n\\end{equation}\nIn equation~(\\ref{eq:coeff_reflect}), $m_1$ and $m_2$ are the indices of refraction in two adjacent layers.  When $m_2$ is imaginary, total reflection occurs and the wave is evanescent in that region and its amplitude decays to zero.  However a region of propagation can exist between two regions of evanescence.  This readily occurs for jets, where the intrinsic phase speed varies enough to allow the hydrostatic term to dominate in some regions and not in others.  The region can also occur through variations of the \\BV\\ frequency.  In Figure~\\ref{fig:realtrapped} we see a wave that is trapped in the relatively quiescent region between $z\/H_p \\approx 1$ and $z\/H_p \\approx 4$.  Outside this region, the value of Re$(m)$ is small, or zero.  Trapped in this region the wave is able to interact with itself and, under appropriate conditions, resonate.\n\nThis is another mechanism via which waves may be filtered out by the flow; and so the waves do not reach high altitudes at which saturation can occur.  However, in this case, a trapped wave does not directly interact with a low level flow that changes its characteristics.  Indeed, between the two reflection layers the wave can propagate horizontally---even in the absence of any flow, using the refractive mechanism described above.  As long as the layers do not allow much leakage, it is possible for a trapped wave to cover large horizontal distances---transporting momentum and heat zonally (east-west direction).\n\n\\subsubsection{Ducting}\\label{subsubsec:ducting}\n\nAs well as being trapped in relatively quiescent regions, it is possible for waves to be trapped in a jet.  As alluded to above, it is possible for such a wave to travel within the region of trapping, which is known as a duct or a waveguide.  In Figure~\\ref{fig:realducted}, a non-hydrostatic wave with $c = 700$~m~s$^{-1}$ is trapped within the jet (located at $\\sim$5~mbar level) in our model atmosphere.  Note the small values of Re$(m)$ outside the jet.\n\nIn this case, the flows are significant. Therefore, we use the full dispersion relation,\\\\\n\\begin{widetext} \n\\begin{equation} \\label{eq:fulldispersion}\n\\omega\\ =\\ \\frac{ku_0 + 2H_{\\rho}k\\left[u_0'+H_{\\rho}\\left(2\\left(k^2+m^2\\right)u_0+u_0''\\right)\\right] \\\\ \\pm 2\\left[H^2_{\\rho}k^2\\left(\\left(1+4H^2_{\\rho}\\left(k^2+m^2\\right)\\right)N^2 +\\left(u_0'+H_{\\rho}u_0''\\right)^2\\right)\\right]^{1\/2}}{1+4H^2_{\\rho}\\left(k^2+m^2\\right)} \\, ,\n\\end{equation}\n\\end{widetext}\nto develop expressions for the group velocities.\n\nHowever, the expressions obtained are large and rather unilluminating.  They can be simplified by assuming that $u_0'$ and $u_0''$ are small.  This is realistic since the shear is of the order of $10^{-3}$~s$^{-1}$ and $u_0''$ of the order $10^{-8}$~m$^{-1}$~s$^{-1}$.  This is small compared with the other terms in the expressions.  This then gives \\begin{subequations}\n  \\begin{eqnarray} \\label{eq:grphorvelflow}\n  u_g &\\ =\\ & u_0+\\frac{N\\left[m^2 + 1\/(4H^2_{\\rho})\\right]}{\\left[k^2 + m^2 + 1\/(4H^2_{\\rho})\\right]^{3\/2}} \\\\\n  w_g &\\ =\\ & \\pm\\frac{N k m}{\\left[k^2 + m^2 + 1\/(4H^2_{\\rho})\\right]^{3\/2}}\t\n  \\end{eqnarray}\n\\end{subequations}\nFrom these expressions, we can see that $u_g$ follows $u_0$ as this is the larger term on the right hand side of equation~(\\ref{eq:grphorvelflow}) in our model atmosphere.  In Figure~\\ref{fig:realducted}, the values of $u_g$ and $w_g$ are shown.  Note that in the center of the duct $w_g$ is very small while $u_g$ is large, so that energy is transported along the flow.  At the the top and bottom of the duct the vertical group velocity increases, while the horizontal group velocity falls.  Therefore, propagation here is nearly vertical.  In Figure~\\ref{fig:realducted}, we show $w_g$ as positive, however this is only for the upward propagation of energy, at the top of the duct the wave is reflected and the vertical component becomes negative.  This keeps the wave within the jet.  The ray path followed by the wave, before the reflection, is shown in Figure~\\ref{fig:realductedpath}.\n\n\\begin{figure}[t]\n  \\epsscale{1.1}\n  \\plotone{Paper1Fig9.eps}\n  \\caption{As in Figure~\\ref{fig:realtrapped}, but with $c = 700$~m~s$^{-1}$ and horizontal wavelength $2\\pi \/ k = 1410$~km.  \nThe horizontal group speed $u_g$ ($-\\, -$), vertical group speed $w_g$ ($\\cdots$) and the real     part of the index of refraction $m$ ($-\\cdot -$) are shown.  The     wave, not shown, is trapped in the upper jet between $z\/H_p \\approx 3$ and     $z\/H_p \\approx\\! 9$, the region where $m$ is real, and does not propagate vertically above this region; it is     however, able to propagate along the jet as the large value of $u_g$ within the jet shows.  The wave is     reflected at the boundaries of this region, providing a possibility for resonance.  \\label{fig:realducted}}\n\\end{figure}\n\n\\begin{figure}\n  \\epsscale{1.1}\n  \\plotone{Paper1Fig10.eps}\n  \\caption{This shows the path of propagation of the wave in Figure~\\ref{fig:realducted} assuming that properties of the duct do not change in the $x$-direction. The path shown is the first crossing of the duct, that is until the wave encounters the top reflection layer. At this point the wave will reflect and then propagate downwards in a mirror image of this path. Note that the wave travels nearly one planetary radius before reflection. This means with just six reflections the wave will have nearly circumnavigated the planet. Of course in the real situation the properties of the duct will change in the $x$-direction and the wave will probably either leak out of the duct or dissipate before the circumnavigation is complete. \\label{fig:realductedpath}}\n\\end{figure}\n\nThe wave can travel large distances in this duct; but, eventually, the wave will either escape the duct or be dissipated. The range of speeds in the jet may change so that a critical level for the wave is created.  The wave will then be reabsorbed into the flow.  Alternatively, if the flow or \\BV\\ frequency outside the jet changes so that the buoyancy term is no longer small and is dominated by the non-hydrostatic term, then the reflection is no longer total and the wave will then leak out of the duct.  This can, for example, happen when propagating into a colder region, assuming the lapse rate remains constant.  As can be seen from equation~(\\ref{eq:BVF}) a fall in temperature with constant lapse rate will cause an increase in the \\BV\\ frequency and thus an increase in the buoyancy term. This may occur very far from the original region of wave excitation. Indeed, in the example given, it is possible to envisage jets ducting waves and so transporting energy from the dayside of a tidally locked planet to the colder nightside where the waves escape the jet and propagate away from the duct before dissipating.\n\n\\section{Implications for Circulation Models}\\label{sec:implications}\n\nThe effects of gravity waves discussed in this work on the larger-scale circulation must be parameterized in global models because the spatial resolution---both horizontal and vertical---required to model them is currently prohibitive.  The waves important to the large-scale extrasolar planet atmospheric circulation have horizontal length scales ranging from approximately $\\sim\\!  10^5$~m to $\\sim\\! 10^7$~m and vertical wavelengths as small as $\\sim\\!  10^4$~m.  Waves with periods of few hours can carry significant momentum and energy flux vertically, but the sources of these waves include processes that are not included or resolvable in current circulation models.\n\nThe difficulty with representing gravity waves in GCMs exists even for the GCMs of the Earth.  For example, the parameterization for convection is not aimed at producing realistic gravity waves \\citep{Collins2004}.  However, not representing gravity waves can affect the accuracy of the GCMs.  The lack of gravity wave drag can lead to the overestimation of wind speeds, resulting in faster and narrower jets than observed \\citep{McLandress1998}.  Further, gravity waves introduce turbulence with subsequent mixing and thermal transport \\citep{Fritts2003}.  This leads to greater homogenization of the atmosphere with a reduction in, for example, temperature gradients.  Gravity waves also interact with planetary waves, playing a role in important transient phenomena (such as sudden stratospheric warming).  In the absence of gravity waves, these phenomena are not accurately modeled \\citep{Richter2010}.\n\nThere are many parametrization schemes currently incorporated or proposed for general circulation modeling \\citep{McLandress1998}.  In all of the schemes, the basic components are 1) specification of the characteristic of the waves at the source level, 2) wave propagation and evolution as a function of altitude, and 3) effects on and by the atmosphere.  All of them are essentially linear and one-dimensional, in that waves only propagate vertically and that only vertical variation in the background influence the propagation.  As seen in this work, linear theory still requires information such as the wave's phase speed $c$ and wavenumber $\\mathbi{k}$, for example. A more complete theory would need spatial and temporal spectral information. Intermittency is another crucial feature that would need to be taken into account.  The primary differences in various schemes pertain to the treatment of nonlinearity and specificity of wave dissipation mechanisms.\n \nCurrently, all global circulation models of hot Jupiters suggest the presence of a low number of zonal jets.  However, all the models do not have the resolution required to adequately resolve gravity waves and are subject to all of the limitations described above.  This issue has been previously raised by \\citet{Cho2008a}, in which they advocate caution against quantitative interpretation of current model results. For example, without the inclusion of the wave effects discussed in this work, high jet speeds and precise eastward shift of putative ``hot spots'' can be questioned \\citep[e.g.,][]{Cooper2005,Knutson2007,Langton2007} \n\n\\section{Conclusion}\\label{sec:conclusion}\n\nGravity wave propagation and momentum and energy deposition are complicated by the environment in which the wave propagates.  For example, spatial variability of the background winds causes the wave to be refracted, reflected, focused, and ducted. Additionally, temporal variability of the background winds cause the wave to alter its phase speed.  Still further complications arise due to the wave's ability to generate turbulence, which can modify the source or serve as a secondary source, and the wave's interaction with the vortical (rotational) mode of the atmosphere.  Many of these issues are as yet not well-understood and are currently areas of active research.\n\nIn this work, we have emphasized only some of these issues.  We have shown that gravity waves propagate and transport momentum and heat in the atmospheres of hot extrasolar planets and that the waves play an important role in the atmosphere.  They modify the circulation through exerting accelerations on the mean flow whenever the wave encounters a critical level or saturates.  They also transport heat to the upper stratosphere and thermosphere, causing significant heating in these regions.  Moreover, through ducting, they also provide a mechanism for transporting heat from the dayside of tidally synchronized planets.\n\nBefore relying on GCMs for quantitative descriptions of hot extrasolar planet atmospheric circulations, further work needs to be performed to ensure that the effects of important sub-scale phenomena, such as the gravity waves discussed here, are accurately parameterized and included in the GCMs.\n\n\\section*{Acknowledgments}\nThe authors thank Heidar Thrastarson for generously sharing information from his simulation work on extrasolar planet atmospheric circulation.  We also thank Orkan Umurhan for helpful discussions and the anonymous referee for helpful suggestions. C.W. is supported by the Science and Technology Facilities Council (STFC). J.Y-K.C. is supported by NASA NNG04GN82G and STFC PP\/E001858\/1 grants.\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nFinite dimensional algebras over a field are classical, well studied\nmathematical objects.  Their representation theory is a particularly\nlarge and active area which has inspired a number of powerful\nmathematical techniques, not least Auslander-Reiten theory which is a\nbeautiful and effective set of tools and ideas.  See Appendix\n\\ref{app:AR} and the references listed there for an introduction.\n\nIt seems reasonable to look for applications of Auslander-Reiten (AR)\ntheory to areas outside representation theory.  Specifically, let $X$\nbe a topological space.  The singular cochain complex $\\operatorname{C}^*(X;k)$\nwith coefficients in a field $k$ of characteristic $0$ is a\nDifferential Graded algebra which has been studied intensively, in\nparticular in rational homotopy theory, see \\cite{FHTbook}.  For an\nintroduction to Differential Graded (DG) homological algebra, see\nAppendix \\ref{app:DG} and the references listed there.  The singular\ncohomology $\\operatorname{H}^*(X;k)$ is defined as the cohomology of the\ncomplex $\\operatorname{C}^*(X;k)$; it is a graded algebra.  Now let $X$ be\nsimply connected with $\\dim_k \\operatorname{H}^*(X;k) < \\infty$; then\n$\\operatorname{C}^*(X;k)$ is quasi-isomorphic to a DG algebra $R$ with $\\dim_k R\n< \\infty$, and it is natural to try to apply AR theory to $R$.  This\nwas the subject of \\cite{artop}, \\cite{arquiv}, and \\cite{Schmidt},\nand the object of this paper is to review the results of those papers.\n\nAmong the highlights is Theorem \\ref{thm:Chain_CY} from which comes\nthe title of the paper.  Consider the derived category of DG\nleft-$R$-modules, $\\sD(R)$, which is equivalent to\n$\\sD(\\operatorname{C}^*(X;k))$ since the two DG algebras are quasi-isomorphic.\nThe latter category has the full subcategory $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X;k))$\nconsisting of compact DG modules; these play the role of finitely\ngenerated representations.  Theorem \\ref{thm:Chain_CY} now says that\nif $k$ has characteristic $0$, then\n\\begin{align}\n\\label{equ:z}\n  & \\mbox{$\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X;k))$ is an $n$-Calabi-Yau category} \\\\\n\\nonumber\n  & \\mbox{$\\Leftrightarrow$ $X$ has $n$-dimensional Poincar\\'{e}\n    duality over $k$.}\n\\end{align}\n\nLet me briefly explain the terminology.  A triangulated category\n$\\mathsf T$, such as for instance $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X;k))$, is called\n$n$-Calabi-Yau if $n$ is the smallest non-negative integer for which\n$\\Sigma^n$, the $n$th power of the suspension functor, is a Serre\nfunctor, that is, permits natural isomorphisms\n\\[\n  \\operatorname{Hom}_k(\\operatorname{Hom}_{\\mathsf T}(M,N),k) \\cong \\operatorname{Hom}_{\\mathsf T}(N,\\Sigma^n M).\n\\]\nThe topological space $X$ is said to have $n$-dimensional Poincar\\'{e}\nduality over $k$ if there is an isomorphism\n\\[\n  \\operatorname{Hom}_k(\\operatorname{H}^*(X;k),k) \\cong \\Sigma^n \\operatorname{H}^*(X;k)\n\\]\nof graded left-$\\operatorname{H}^*(X;k)$-modules.\n\nExamples of $n$-Calabi-Yau categories are higher cluster categories,\nsee \\cite[sec.\\ 4]{KellerReiten}, and examples of spaces with\n$n$-dimensional Poincar\\'{e} duality are compact $n$-dimensional\nmanifolds.  Equation \\eqref{equ:z} provides a link between the\ncurrently po\\-pu\\-lar theory of Calabi-Yau categories and algebraic\ntopology.  It also gives a new class of examples of Calabi-Yau\ncategories which, so far, typically have been exemplified by higher\ncluster categories.  The new categories appear to behave very\ndifferently from higher cluster categories, cf.\\ Section\n\\ref{sec:open}, Problem \\ref{prob:CY}.\n\nA number of other results are also obtained, not least on the\nstructure of the AR quiver of $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X;k))$ which, for a\nspace with Poincar\\'{e} duality, consists of copies of the repetitive\nquiver ${\\mathbb Z} A_{\\infty}$, see Theorem \\ref{thm:Chain_Gamma}.\n\nIn a speculative vein, the theory presented here ties in with the\nversion of non-commutative geometry in which a DG algebra, or more\ngenerally a DG category, is viewed as a non-commutative scheme.  The\nidea is to think of the derived category of the DG algebra or DG\ncategory as being the derived category of quasi-coherent sheaves on a\nnon-commutative scheme (which does not actually exist).  There appear\nso far to be no published references for this viewpoint which has been\nbrought forward by Drinfeld and Kontsevich, but it does seem to call\nfor a detailed study of the derived categories of DG algebras and DG\ncategories.  Auslander-Reiten theory is an obvious tool to try, and\n\\cite{artop}, \\cite{arquiv}, and \\cite{Schmidt} along with this paper\ncan, perhaps, be viewed as a first, modest step.\n\nAs indicated, the paper is a review.  The results were known\npreviously, the main references being \\cite{artop}, \\cite{arquiv}, and\n\\cite{Schmidt}; more details of the origin of individual results are\ngiven in the introductions to the sections.  There is no claim to\no\\-ri\\-gi\\-na\\-li\\-ty, except that some of the proofs are new.  It is\nalso the first time this material has appeared together.\n\nMost of the paper is phrased in terms of the DG algebra $R$ rather\nthan $\\operatorname{C}^*(X;k)$, see Setup \\ref{set:blanket}.  This is merely a\nnotational convenience: $R$ and $\\operatorname{C}^*(X;k)$ are quasi-isomorphic,\nso have equivalent derived categories.  Hence, all results about the\nderived category of $R$ also hold for the derived category of\n$\\operatorname{C}^*(X;k)$.  The paper is organized as follows:\n\nSome background on DG homological algebra and AR theory is collected\nin two appendices, \\ref{app:DG} and \\ref{app:AR}.\n\nSection \\ref{sec:cochain} gives some preliminary results on cochain DG\nalgebras and their DG modules.  The main result is Theorem\n\\ref{thm:Gorenstein} which gives a number of alternative descriptions\nof when $R$ is a so-called Gorenstein DG algebra.  The importance of\nthis condition is that $\\operatorname{C}^*(X;k)$ is Gorenstein precisely when\n$X$ has Poincar\\'{e} duality.\n\nSection \\ref{sec:AR} studies the existence of AR\ntriangles in the category $\\sD^{\\operatorname{c}}(R)$, which turns out to be\nequivalent to $R$ being Gorenstein by Theorem \\ref{thm:R}.\n\nSection \\ref{sec:local} considers the local structure of the AR quiver\n$\\Gamma$ of $\\sD^{\\operatorname{c}}(R)$.  If $R$ is Gorenstein with $\\dim_k\n\\operatorname{H}\\!R \\geq 2$, then Theorem \\ref{thm:quiver_local_structure}\nshows that each component of $\\Gamma$ is isomorphic to ${\\mathbb Z}\nA_{\\infty}$.\n\nSection \\ref{sec:global} reports on work by Karsten Schmidt.  It looks\nat the global structure of $\\Gamma$ where the results are so far less\nconclusive.  If $\\dim_k \\operatorname{H}\\!R = 2$, then $\\Gamma$ has precisely\n$d-1$ components isomorphic to ${\\mathbb Z} A_{\\infty}$, where $d = \\sup \\{\\,\ni \\,|\\, \\operatorname{H}^i\\!R \\neq 0 \\,\\}$.  On the other hand, if $R$ is\nGorenstein with $\\dim_k \\operatorname{H}\\!R \\geq 3$, then $\\Gamma$ has\ninfinitely many components, and if $\\dim_k \\operatorname{H}^e\\!R \\geq 2$ for\nsome $e$, then it is even possible to find families of distinct\ncomponents indexed by projective manifolds, and these manifolds can be\nof arbitrarily high dimension.\n\nSection \\ref{sec:topology} makes explicit the highlights of the theory\nfor the algebras $\\operatorname{C}^*(X;k)$.\n\nSection \\ref{sec:open} is a list of open problems.\n\n\\medskip\n\\noindent\n{\\bf Acknowledgement.}\nSome of the results of this paper, not least the ones of Section\n\\ref{sec:global}, are due to Karsten Schmidt.  I thank him for a\nnumber of communications on his work, culminating in \\cite{Schmidt}.\nI thank Henning Krause, Andrzej Skowronski, and the referee for\ncomments to a previous version of the paper.  I am grateful to Andrzej\nSkowronski for the very succesful organization of ICRA XII in Torun,\nAugust 2007, and for inviting me to submit this paper to the ensuing\nvolume ``Trends in Representation Theory of Algebras and Related\nTopics''.\n\n\n\n\n\\section{Cochain Differential Graded algebras}\n\\label{sec:cochain}\n\n\nThis section provides some results on cochain Differential Graded (DG)\nalgebras, not least on the ones which are Gorenstein.  The results\nfirst appeared in \\cite{artop}, except Lemma \\ref{lem:inf} which is\n\\cite[lem.\\ 1.5]{FJcochain} and Theorem \\ref{thm:k} which is\n\\cite[cor.\\ 3.12]{Schmidt}.\n\nFor background and terminology on DG algebras and their derived\ncategories, see Appendix \\ref{app:DG}.\n\n\\begin{setup}\n\\label{set:blanket}\nIn Sections \\ref{sec:cochain} through \\ref{sec:open}, $k$ is a\nfield and $R$ is a DG algebra over $k$ which has the form \n\\[\n  \\cdots \\rightarrow 0 \\rightarrow k \\rightarrow 0 \\rightarrow R^2\n  \\rightarrow R^3 \\rightarrow \\cdots,\n\\]\nthat is, $R^{<0} = 0$, $R^0 = k$, and $R^1 = 0$.  It will be assumed\nthat $\\dim_k R < \\infty$, and throughout,\n\\[\n  d = \\sup R\n\\]\nwhere $\\sup$ is as in Definition \\ref{def:sup_and_inf}.\n\\end{setup}\n\nNote that either $d = 0$, in which case $R$ is quasi-isomorphic to\n$k$, or $d \\geq 2$.\n\n\\begin{remark}\nIf $X$ is a simply connected topological space with $\\dim_k\n\\operatorname{H}^*(X;k) < \\infty$ and $k$ has characteristic $0$, then\n$\\operatorname{C}^*(X;k)$ is quasi-isomorphic to a DG algebra $R$ satisfying the\nconditions of Setup \\ref{set:blanket} by \\cite[exa.\\ 6, p.\\ \n146]{FHTbook}.  This means that the derived categories of\n$\\operatorname{C}^*(X;k)$ and $R$ are equivalent, and hence, all results about\nthe derived category of $R$ also hold for the derived category of\n$\\operatorname{C}^*(X;k)$.\n\nThe highlights of the theory will be made explicit for $\\operatorname{C}^*(X;k)$\nin Section \\ref{sec:topology}.\n\\end{remark}\n\n\\begin{proposition}\n\\label{pro:DcinDf}\nThe full subcategory $\\sD^{\\operatorname{c}}(R)$ of compact objects of the derived\ncategory $\\sD(R)$ is contained in $\\sD^{\\operatorname{f}}(R)$, the full subcategory\nof $\\sD(R)$ of objects with $\\dim_k \\operatorname{H}\\!M < \\infty$.\n\\end{proposition}\n\n\\begin{proof}\nThe DG module ${}_{R}R$ is in $\\sD^{\\operatorname{f}}(R)$ by assumption, and\n$\\sD^{\\operatorname{c}}(R)$ consists of the DG modules which are finitely built from\nit, cf.\\ Definition \\ref{def:D}, so it follows that $\\sD^{\\operatorname{c}}(R)$ is\ncontained in $\\sD^{\\operatorname{f}}(R)$.\n\\end{proof}\n\n\\begin{proposition}\n\\label{pro:DcandDfKrullSchmidt}\nThe triangulated categories $\\sD^{\\operatorname{f}}(R)$ and $\\sD^{\\operatorname{c}}(R)$ have finite\ndimensional Hom spaces and split idempotents.\n\nConsequently, $\\sD^{\\operatorname{f}}(R)$ and $\\sD^{\\operatorname{c}}(R)$ are Krull-Schmidt\ncategories. \n\\end{proposition}\n\n\\begin{proof}\nIf $M$ is in $\\sD^{\\operatorname{f}}(R)$, then it is finitely built from ${}_{R}k$ in\n$\\sD(R)$, see Remark \\ref{rmk:Df}.  So to see that $\\sD^{\\operatorname{f}}(R)$\nhas finite dimensional Hom spaces, it is enough to see that\n$\\operatorname{Hom}_{\\sD(R)}(\\Sigma^i k,k)$ is finite dimensional for each $i$,\nwhere $\\Sigma$ denotes the suspension functor of $\\sD(R)$.\n\nLet $F$ be a minimal semi-free resolution of ${}_{R}(\\Sigma^i k)$;\nthen\n\\begin{align*}\n  \\operatorname{Hom}_{\\sD(R)}(\\Sigma^i k,k)\n  & \\stackrel{\\rm (a)}{\\cong} \\operatorname{H}^0 \\operatorname{RHom}_R(\\Sigma^i k,k) \\\\\n  & \\stackrel{\\rm (b)}{\\cong} \\operatorname{H}^0 \\operatorname{Hom}_R(F,k) \\\\\n  & \\stackrel{\\rm (c)}{\\cong} \\operatorname{Hom}_{R^{\\natural}}(F^{\\natural},k^{\\natural})^0\\\\\n  & = (*)\n\\end{align*}\nwhere (a) is by Definition \\ref{def:Hom_and_Tensor} and (b) and (c)\nare by Lemma \\ref{lem:semi-free}, (2) and (5).  However, Lemma\n\\ref{lem:semi-free}(3) says that $F^{\\natural} \\cong \\bigoplus_{j \\leq\ni} \\Sigma^j(R^{\\natural})^{(\\beta_j)}$ with the $\\beta_j$ finite,\nwhere superscript $(\\beta)$ indicates the direct sum of $\\beta$ copies\nof the module, and so\n\\[\n  (*) \\cong k^{(\\beta_0)}.\n\\]\nThis is finite dimensional.\n\nSince $\\sD^{\\operatorname{c}}(R)$ is contained in $\\sD^{\\operatorname{f}}(R)$ by Proposition\n\\ref{pro:DcinDf}, it follows that $\\sD^{\\operatorname{c}}(R)$ also has finite\ndimensional Hom spaces.\n\nIdempotents split in both $\\sD^{\\operatorname{f}}(R)$ and $\\sD^{\\operatorname{c}}(R)$ since\nby \\cite[prop.\\ 3.2]{BN} they split already in $\\sD(R)$ because this\nis a triangulated category with set indexed co\\-pro\\-ducts.\n\nBy \\cite[p.\\ 52]{Ringel}, both $\\sD^{\\operatorname{f}}(R)$ and $\\sD^{\\operatorname{c}}(R)$ are\nKrull-Schmidt categories.\n\\end{proof}\n\nRecall from Definition \\ref{def:sup_and_inf} the notion of $\\inf$ of a\nDG module, and from Definition \\ref{def:DG} that $R^{\\operatorname{o}}$ is the\nopposite DG algebra of $R$ and that DG left-$R^{\\operatorname{o}}$-modules can be\nviewed as DG right-$R$-modules.\n\n\\begin{lemma}\n\\label{lem:inf}\nLet $M$ be in $\\sD^{\\operatorname{f}}(R^{\\operatorname{o}})$ and let $N$ be in $\\sD^{\\operatorname{f}}(R)$.  Then \n\\[\n  \\inf(M \\stackrel{\\operatorname{L}}{\\otimes}_R N) = \\inf M + \\inf N.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nIf $M$ or $N$ is isomorphic to zero then the equation reads $\\infty\n= \\infty$, so let me assume not.  Then $i = \\inf M$ and $j = \\inf N$\nare integers.\n\nLemma \\ref{lem:truncations}(1) says that $M$ can be replaced with a\nquasi-isomorphic DG mo\\-du\\-le which satisfies $M^{\\ell} = 0$ for\n$\\ell < i$.\n\nLemma \\ref{lem:semi-free}(3) says that $N$ has a semi-free resolution\n$F$ which satisfies that $F^{\\natural} \\cong \\bigoplus_{\\ell \\leq -j}\n\\Sigma^{\\ell} (R^{\\natural})^{(\\beta_{\\ell})}$, and it follows that\n$(M \\otimes_R F)^{\\natural} \\cong \\bigoplus_{\\ell \\leq -j}\n\\Sigma^{\\ell} (M^{\\natural})^{(\\beta_{\\ell})}$.\n\nSince $M^{\\ell} = 0$ for $\\ell < i$, this implies that $(M \\otimes_R\nF)^{\\ell} = 0$ for $\\ell < i+j$.  In particular, $\\inf(M \\otimes_R F)\n\\geq i+j$ whence\n\\begin{equation}\n\\label{equ:h}\n  \\inf(M \\stackrel{\\operatorname{L}}{\\otimes}_R N) \\geq i+j = \\inf M + \\inf N.\n\\end{equation}\n\nConversely, to give a morphism of DG left-$R$-modules $\\Sigma^{-j}R\n\\rightarrow N$ is the same thing as to give the image $z$ of\n$\\Sigma^{-j}(1_R)$, and $z$ is a cycle in $N^j$.  Since\n$\\operatorname{H}^j(\\Sigma^{-j}R) \\cong \\operatorname{H}^0\\!R \\cong k$, the induced\nmap $\\operatorname{H}^j(\\Sigma^{-j}R) \\rightarrow \\operatorname{H}^j\\!N$ is just the\nmap $k \\rightarrow \\operatorname{H}^j\\!N$ sending $1_k$ to the cohomology\nclass of $z$.  Hence, picking cycles $z_{\\alpha}$ whose cohomology\nclasses form a $k$-basis of $\\operatorname{H}^j\\!N$ and constructing a\nmorphism $\\Sigma^{-j}R^{(\\beta)} \\rightarrow N$ by sending the\nelements $\\Sigma^{-j}(1_R)$ to the $z_{\\alpha}$ gives that the induced\nmap $\\operatorname{H}^j(\\Sigma^{-j}R^{(\\beta)}) \\rightarrow \\operatorname{H}^j\\!N$ is\nan isomorphism.  Complete to a distinguished triangle\n\\begin{equation}\n\\label{equ:i}\n  \\Sigma^{-j}R^{(\\beta)} \\rightarrow N \\rightarrow N^{\\prime\\prime} \\rightarrow;\n\\end{equation}\nsince $\\operatorname{H}^{j+1}(\\Sigma^{-j}R^{(\\beta)}) \\cong\n\\operatorname{H}^1(R^{(\\beta)}) = 0$, the long exact cohomology sequence\nshows\n\\begin{equation}\n\\label{equ:k}\n  \\inf N^{\\prime\\prime} \\geq j+1.\n\\end{equation}\n\nTensoring the distinguished triangle \\eqref{equ:i} with\n$M$ gives \n\\[\n  \\Sigma^{-j}M^{(\\beta)} \\rightarrow M \\stackrel{\\operatorname{L}}{\\otimes}_R N\n    \\rightarrow M \\stackrel{\\operatorname{L}}{\\otimes}_R N^{\\prime\\prime} \\rightarrow\n\\]\nand the long exact cohomology sequence of this contains\n\\begin{equation}\n\\label{equ:l}\n  \\operatorname{H}^{i+j-1}(M \\stackrel{\\operatorname{L}}{\\otimes}_R N^{\\prime\\prime})\n    \\rightarrow \\operatorname{H}^{i+j}(\\Sigma^{-j}M^{(\\beta)})\n    \\rightarrow \\operatorname{H}^{i+j}(M \\stackrel{\\operatorname{L}}{\\otimes}_R N).\n\\end{equation}\nThe inequality \\eqref{equ:h} can be applied to $M$ and $N^{\\prime\\prime}$;\nbecause of the inequality \\eqref{equ:k}, this gives $\\inf(M \\stackrel{\\operatorname{L}}{\\otimes}_R\nN^{\\prime\\prime}) \\geq i+j+1$ so the first term of the exact sequence\n\\eqref{equ:l} is zero.  The second term is\n$\\operatorname{H}^{i+j}(\\Sigma^{-j}M^{(\\beta)}) \\cong\n\\operatorname{H}^i(M^{(\\beta)})$ which is non-zero since $i = \\inf M$.  It\nfollows that the third term is non-zero, so\n\\[\n  \\inf(M \\stackrel{\\operatorname{L}}{\\otimes}_R N) \\leq i+j = \\inf M + \\inf N.\n\\]\nCombining with the inequality \\eqref{equ:h} proves the lemma.\n\\end{proof}\n\n\\begin{definition}\nThe DG algebra $R$ is said to be Gorenstein if it satisfies the\nequivalent conditions of the following theorem.\n\\end{definition}\n\nIn the theorem, recall from Definition \\ref{def:dual} that $\\operatorname{D}(-) =\n\\operatorname{Hom}_k(-,k)$.\n\n\\begin{theorem}\n\\label{thm:Gorenstein}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\n  \\item  There are isomorphisms of $k$-vector spaces\n\\[\n  \\operatorname{Ext}_R^i(k,R) \\; \\cong \\;\n  \\left\\{\n    \\begin{array}{cl}\n      k & \\mbox{for $i = d$}, \\\\\n      0 & \\mbox{otherwise}\n    \\end{array}\n  \\right\\} \\; \\cong \\;\n  \\operatorname{Ext}_{R^{\\operatorname{o}}}^i(k,R).\n\\]\n\n  \\item  There are isomorphisms of graded $\\operatorname{H}\\!R$-modules\n\\[\n  \\mbox{${}_{\\operatorname{H}\\!R}(\\operatorname{D}\\!\\operatorname{H}\\!R) \\cong {}_{\\operatorname{H}\\!R}(\\Sigma^d \\operatorname{H}\\!R)$\n        \\;\\;and\\;\\;\n        $(\\operatorname{D}\\!\\operatorname{H}\\!R)_{\\operatorname{H}\\!R} \\cong (\\Sigma^d \\operatorname{H}\\!R)_{\\operatorname{H}\\!R}$.}\n\\]\n\n  \\item  There are isomorphisms\n\\[\n  \\mbox{${}_{R}(\\operatorname{D}\\!R) \\cong {}_{R}(\\Sigma^d R)$ \\;in\\; $\\sD(R)$\n        \\;\\;and\\;\\;\n        $(\\operatorname{D}\\!R)_R \\cong (\\Sigma^d R)_R$ \\;in\\; $\\sD(R^{\\operatorname{o}})$.}\n\\]\n\n  \\item  $\\dim_k \\operatorname{Ext}_R(k,R) < \\infty$ and $\\dim_k\n    \\operatorname{Ext}_{R^{\\operatorname{o}}}(k,R) < \\infty$. \n\n  \\item  ${}_{R}(\\operatorname{D}\\!R)$ is in $\\sD^{\\operatorname{c}}(R)$ and $(\\operatorname{D}\\!R)_R$ is in\n    $\\sD^{\\operatorname{c}}(R^{\\operatorname{o}})$. \n\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n(1)$\\Rightarrow$(3).  Let $F$ be a minimal semi-free resolution of\n${}_{R}(\\operatorname{D}\\!R)$.  Then\n\\begin{equation}\n\\label{equ:d}\n  \\operatorname{Ext}_{R^{\\operatorname{o}}}(k,R)\n  \\stackrel{\\rm (a)}{\\cong} \\operatorname{Ext}_R(\\operatorname{D}\\!R,k)\n  = \\operatorname{H}(\\operatorname{RHom}_R(\\operatorname{D}\\!R,k))\n  \\stackrel{\\rm (b)}{\\cong} \\operatorname{Hom}_{R^{\\natural}}(F^{\\natural},k^{\\natural})\n\\end{equation}\nwhere (a) is by duality and (b) is by Lemma \\ref{lem:semi-free}, (2)\nand (5).  If the second isomorphism in (1) holds, then this implies\n$F^{\\natural} \\cong \\Sigma^d R^{\\natural}$.  But then there is clearly\nonly a single step in the semi-free filtration of $F$, whence $F \\cong\n{}_{R}(\\Sigma^d R)$ so ${}_{R}(\\operatorname{D}\\!R) \\cong {}_{R}(\\Sigma^d R)$,\nproving the first isomorphism in (3).  Likewise, the first\nisomorphism in (1) implies the second isomorphism in (3).\n\n(3)$\\Rightarrow$(2).  This follows by taking cohomology.\n\n(2)$\\Rightarrow$(1).  This follows from the Eilenberg-Moore spectral\nsequence\n\\[\n  E_2^{pq} = \\operatorname{Ext}_{\\operatorname{H}\\!R}^p(k,\\operatorname{H}\\!R)^q\n  \\Rightarrow \\operatorname{Ext}_R^{p+q}(k,R)\n\\]\nwhich exists by \\cite[1.3(2)]{FHTpaper}, and the corresponding\nspectral sequence over $R^{\\operatorname{o}}$.\n\n(4)$\\Leftrightarrow$(5).  Lemma \\ref{lem:semi-free}(3) says that the\nsemi-free resolution $F$ of ${}_{R}(\\operatorname{D}\\!R)$ has $F^{\\natural} \\cong\n\\bigoplus_i \\Sigma^i(R^{\\natural})^{(\\beta_i)}$, and Equation\n\\eqref{equ:d} shows that $\\dim_k \\operatorname{Ext}_{R^{\\operatorname{o}}}(k,R)$ is the\nnumber of direct summands $\\Sigma^i R^{\\natural}$.  By Lemma\n\\ref{lem:semi-free}(4), this number is finite if and only if\n${}_{R}(\\operatorname{D}\\!R)$ is in $\\sD^{\\operatorname{c}}(R)$, so the second condition in (4)\nis equivalent to the first condition in (5) and vice versa.\n\n(1)$\\Rightarrow$(4) is clear.\n\n(4)$\\Rightarrow$(1).  When (4) holds, so does (5) by the previous part\nof the proof; hence ${}_{R}(\\operatorname{D}\\!R)$ is finitely built from\n${}_{R}R$.  Then the canonical morphism\n\\[\n  \\operatorname{RHom}_R(k,R) \\stackrel{\\operatorname{L}}{\\otimes}_R \\operatorname{D}\\!R\n  \\rightarrow \\operatorname{RHom}_R(k,R \\stackrel{\\operatorname{L}}{\\otimes}_R \\operatorname{D}\\!R)\n\\]\nis an isomorphism, because it clearly is if $\\operatorname{D}\\!R$ is replaced\nwith $R$.  That is,\n\\begin{equation}\n\\label{equ:c}\n  \\operatorname{RHom}_R(k,R) \\stackrel{\\operatorname{L}}{\\otimes}_R \\operatorname{D}\\!R \\cong k.\n\\end{equation}\nSince (4) holds, $\\operatorname{RHom}_R(k,R)$ is in $\\sD^{\\operatorname{f}}(R^{\\operatorname{o}})$, so Lemma\n\\ref{lem:inf} applies to the tensor product and gives\n\\[\n  \\inf \\operatorname{RHom}_R(k,R) + \\inf \\operatorname{D}\\!R = \\inf k = 0\n\\]\nwhich amounts to\n\\[\n  \\inf \\operatorname{RHom}_R(k,R) = d.\n\\]\n\nOn the other hand, adjointness gives the first of the next isomorphisms,\n\\[\n  \\operatorname{RHom}_k((\\operatorname{D}\\!R) \\stackrel{\\operatorname{L}}{\\otimes}_R k,k)\n  \\cong \\operatorname{RHom}_R(k,\\operatorname{RHom}_k(\\operatorname{D}\\!R,k))\n  \\cong \\operatorname{RHom}_R(k,R),\n\\]\nand so\n\\begin{align*}\n  \\sup \\operatorname{RHom}_R(k,R)\n  & = \\sup \\operatorname{RHom}_k((\\operatorname{D}\\!R) \\stackrel{\\operatorname{L}}{\\otimes}_R k,k) \\\\\n  & = - \\inf((\\operatorname{D}\\!R) \\stackrel{\\operatorname{L}}{\\otimes}_R k) \\\\\n  & \\stackrel{\\rm (c)}{=} - \\inf \\operatorname{D}\\!R - \\inf k \\\\\n  & = d\n\\end{align*}\nwhere (c) is by Lemma \\ref{lem:inf} again.  Hence the cohomology of\n$\\operatorname{RHom}_R(k,R)$ is concentrated in degree $d$, and it is not hard to\nshow that hence\n\\[\n  \\operatorname{RHom}_R(k,R) \\cong (\\Sigma^{-d}k^{(\\beta)})_R\n\\]\nfor some $\\beta$.  Inserting this into Equation \\eqref{equ:c} shows\n$\\beta = 1$, so\n\\[\n  \\operatorname{RHom}_R(k,R) \\cong (\\Sigma^{-d}k)_R.\n\\]\nThis is equivalent to the first isomorphism in (1), and the second one\nfollows by a symmetric argument.\n\\end{proof}\n\n\\begin{theorem}\n\\label{thm:k}\nIf $\\dim_k \\operatorname{H}\\!R \\geq 2$, then ${}_{R}k$ is not in $\\sD^{\\operatorname{c}}(R)$.\n\\end{theorem}\n\n\\begin{proof}\nRecall from Definition \\ref{def:sup_and_inf} the notion of amplitude\nof a DG module.  There is an amplitude inequality $\\operatorname{amp}(M \\stackrel{\\operatorname{L}}{\\otimes}_R\nN) \\geq \\operatorname{amp} M$ for $M$ in $\\sD^{\\operatorname{f}}(R^{\\operatorname{o}})$ and $N$ in $\\sD^{\\operatorname{c}}(R)$.\nThis was first stated in \\cite[prop.\\ 3.11]{Schmidt}; see\n\\cite[cor.\\ 4.4]{FJcochain} for an alternative proof.\n\nIf ${}_{R}k$ were in $\\sD^{\\operatorname{c}}(R)$, then this would give $\\operatorname{amp}(R\n\\stackrel{\\operatorname{L}}{\\otimes}_R k) \\geq \\operatorname{amp} R$, that is, $0 \\geq \\operatorname{amp} R$ contradicting\n$\\dim_k \\operatorname{H}\\!R \\geq 2$ whereby $R$ must (also) have cohomology in\na degree different from $0$.\n\\end{proof}\n\n\n\n\n\\section{Auslander-Reiten triangles over Differential \\\\ Gra\\-ded algebras}\n\\label{sec:AR}\n\n\nIn this section, it is proved that the compact derived category\n$\\sD^{\\operatorname{c}}(R)$ has Auslander-Reiten (AR) triangles if and only if $R$ is\na Gorenstein DG algebra.  In this case, a formula is found for the AR\ntranslation of $\\sD^{\\operatorname{c}}(R)$.  These results first appeared in\n\\cite{artop}.\n\nFor background on AR theory, see Appendix \\ref{app:AR}.\n\nIn the following proposition, note that $\\operatorname{D}\\!R \\stackrel{\\operatorname{L}}{\\otimes}_R P$\ninherits a left-$R$-structure from the DG bi-$R$-module $\\operatorname{D}\\!R$ so\n$\\operatorname{D}\\!R \\stackrel{\\operatorname{L}}{\\otimes}_R P$ is in $\\sD(R)$; see Definition\n\\ref{def:Hom_and_Tensor}.\n\n\\begin{proposition}\n\\label{pro:AR}\nLet $P$ be an indecomposable object of $\\sD^{\\operatorname{c}}(R)$.  There is an\nAR triangle in $\\sD^{\\operatorname{f}}(R)$,\n\\[\n  \\Sigma^{-1}(\\operatorname{D}\\!R \\stackrel{\\operatorname{L}}{\\otimes}_R P) \\rightarrow N \\rightarrow P \\rightarrow.\n\\]\n\\end{proposition}\n\n\\begin{proof}\nSince $P$ is finitely built from ${}_{R}R$, there is a natural\nequivalence\n\\begin{equation}\n\\label{equ:Serre}\n  \\operatorname{D}(\\operatorname{Hom}_{\\sD(R)}(P,-)) \\simeq \\operatorname{Hom}_{\\sD(R)}(-,\\operatorname{D}\\!R \\stackrel{\\operatorname{L}}{\\otimes}_R P),\n\\end{equation}\nsince there is clearly such an equivalence if $P$ is replaced with\n${}_{R}R$.  By \\cite[prop.\\ 4.2]{KrauseARZ}, this means that the AR\ntriangle of the present proposition exists in $\\sD(R)$.\n\nTo complete the proof, observe that the triangle is in fact in\n$\\sD^{\\operatorname{f}}(R)$: The object $P$ is in $\\sD^{\\operatorname{c}}(R)$, so it is in $\\sD^{\\operatorname{f}}(R)$\nby Proposition \\ref{pro:DcinDf}.  Since $R_R$ is in\n$\\sD^{\\operatorname{f}}(R^{\\operatorname{o}})$, the dual ${}_{R}(\\operatorname{D}\\!R)$ is in $\\sD^{\\operatorname{f}}(R)$, and\nsince $P$ is finitely built from $R$, it follows that $\\operatorname{D}\\!R\n\\stackrel{\\operatorname{L}}{\\otimes}_R P$ is also in $\\sD^{\\operatorname{f}}(R)$.  Finally, $N$ is in $\\sD^{\\operatorname{f}}(R)$\nby the long exact cohomology sequence.\n\\end{proof}\n\n\\begin{proposition}\n\\label{pro:AR_triangles_preserved}\nAn AR triangle in $\\sD^{\\operatorname{c}}(R)$ is also an AR triangle in $\\sD^{\\operatorname{f}}(R)$.\n\\end{proposition}\n\n\\begin{proof}\nBy \\cite[lem.\\ 4.3]{KrauseARZ}, each object in $\\sD^{\\operatorname{c}}(R)$ is a pure\ninjective object of $\\sD(R)$.  Hence by \\cite[prop.\\\n3.2]{KrauseARZ}, each AR triangle in $\\sD^{\\operatorname{c}}(R)$ is an AR triangle in\n$\\sD(R)$, and in particular in $\\sD^{\\operatorname{f}}(R)$.\n\\end{proof}\n\n\\begin{proposition}\n\\label{pro:AR2}\n\\begin{enumerate}\n\n  \\item  $\\sD^{\\operatorname{c}}(R)$ has right AR triangles if and only if\n    ${}_{R}(\\operatorname{D}\\!R)$ is in $\\sD^{\\operatorname{c}}(R)$. \n\n  \\item  $\\sD^{\\operatorname{c}}(R)$ has left AR triangles if and only if $(\\operatorname{D}\\!R)_R$\n    is in $\\sD^{\\operatorname{c}}(R^{\\operatorname{o}})$.\n\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n(1).  Suppose that $\\sD^{\\operatorname{c}}(R)$ has right AR triangles.  The object\n${}_{R}R$ of $\\sD^{\\operatorname{c}}(R)$ has endomorphism ring $k$ which is local, so\nthere is an AR triangle $M \\rightarrow N \\rightarrow {}_{R}R\n\\rightarrow$ in $\\sD^{\\operatorname{c}}(R)$.  By Proposition\n\\ref{pro:AR_triangles_preserved}, it is even an AR triangle in\n$\\sD^{\\operatorname{f}}(R)$.  On the other hand, Proposition \\ref{pro:AR} gives that\nthere is also an AR triangle $\\Sigma^{-1}({}_{R}(\\operatorname{D}\\!R))\n\\rightarrow N^{\\prime} \\rightarrow {}_{R}R \\rightarrow$ in\n$\\sD^{\\operatorname{f}}(R)$, and since the right hand terms of the two AR triangles\nare isomorphic, so are the left hand terms, $M \\cong\n\\Sigma^{-1}({}_{R}(\\operatorname{D}\\!R))$.  But $M$ is in $\\sD^{\\operatorname{c}}(R)$, so it\nfollows that $\\Sigma^{-1}({}_{R}(\\operatorname{D}\\!R))$ and hence\n${}_{R}(\\operatorname{D}\\!R)$ is in $\\sD^{\\operatorname{c}}(R)$.\n\nConversely, suppose that ${}_{R}(\\operatorname{D}\\!R)$ is in $\\sD^{\\operatorname{c}}(R)$.  Given\n$P$ in $\\sD^{\\operatorname{c}}(R)$, Proposition \\ref{pro:AR} gives an AR triangle\n\\[\n  \\Sigma^{-1}(\\operatorname{D}\\!R \\stackrel{\\operatorname{L}}{\\otimes}_R P)\n  \\rightarrow N\n  \\rightarrow P\n  \\rightarrow\n\\]\nin $\\sD^{\\operatorname{f}}(R)$.  Since ${}_{R}(\\operatorname{D}\\!R)$ is in $\\sD^{\\operatorname{c}}(R)$, it is\nfinitely built from ${}_{R}R$.  The same is true for $P$, and so\n$\\operatorname{D}\\!R \\stackrel{\\operatorname{L}}{\\otimes}_R P$ is also finitely built from ${}_{R}R$, that\nis, it is in $\\sD^{\\operatorname{c}}(R)$.  It follows that both outer terms of the AR\ntriangle are in $\\sD^{\\operatorname{c}}(R)$, and then so is $N$.  That is, the AR\ntriangle is in $\\sD^{\\operatorname{c}}(R)$, so it is an AR triangle in that category.\n\n(2).  The functors $\\operatorname{RHom}_R(-,R)$ and $\\operatorname{RHom}_{R^{\\operatorname{o}}}(-,R)$ are\nquasi-inverse dualities between $\\sD^{\\operatorname{c}}(R)$ and $\\sD^{\\operatorname{c}}(R^{\\operatorname{o}})$,\nso $\\sD^{\\operatorname{c}}(R)$ has left AR triangles if and only if\n$\\sD^{\\operatorname{c}}(R^{\\operatorname{o}})$ has right AR triangles.  By the right module\nversion of part (1), this happens if and only if $(\\operatorname{D}\\!R)_R$ is in\n$\\sD^{\\operatorname{c}}(R^{\\operatorname{o}})$.\n\\end{proof}\n\n\\begin{theorem}\n\\label{thm:R}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\n  \\item  $\\sD^{\\operatorname{c}}(R)$ has AR triangles.\n\n  \\item  $\\sD^{\\operatorname{c}}(R^{\\operatorname{o}})$ has AR triangles.\n\n  \\item  $R$ is Gorenstein.\n\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nBy Theorem \\ref{thm:Gorenstein}(5), condition (3) is equivalent to\nhaving that ${}_{R}(\\operatorname{D}\\!R)$ is in $\\sD^{\\operatorname{c}}(R)$ and $(\\operatorname{D}\\!R)_R$ is\nin $\\sD^{\\operatorname{c}}(R^{\\operatorname{o}})$.  This is equivalent to condition (1) by\nProposition \\ref{pro:AR2}, and it is equivalent to condition (2) by\nthe right module version of Proposition \\ref{pro:AR2}.\n\\end{proof}\n\n\\begin{remark}\n\\label{rmk:tau}\nAssume the situation of Theorem \\ref{thm:R}.\n\nSince $\\sD^{\\operatorname{c}}(R)$ has AR triangles, \\cite[thm.\\ 4.4]{KrauseARZ} and\nEquation \\eqref{equ:Serre} imply that\n\\begin{equation}\n\\label{equ:Serre2}\n  S(-) = \\operatorname{D}\\!R \\stackrel{\\operatorname{L}}{\\otimes}_R -\n\\end{equation}\nis a Serre functor of $\\sD^{\\operatorname{c}}(R)$, cf.\\ Definition \\ref{def:Serre}.\nSo the AR translation $\\tau$ of $\\sD^{\\operatorname{c}}(R)$ extends to the\nautoequivalence\n\\begin{equation}\n\\label{equ:m}\n  \\Sigma^{-1}(\\operatorname{D}\\!R \\stackrel{\\operatorname{L}}{\\otimes}_R -)\n\\end{equation}\nof $\\sD^{\\operatorname{c}}(R)$, cf.\\ Theorem \\ref{thm:Serre}.  A quasi-inverse\nequivalence is \n\\[\n  \\Sigma \\operatorname{RHom}_{R^{\\operatorname{o}}}(\\operatorname{D}\\!R,R) \\stackrel{\\operatorname{L}}{\\otimes}_R -;\n\\]\nthese two expressions can also be viewed as quasi-inverse\nautoequivalences of $\\sD(R)$.\n\nIf $X$ is an indecomposable object of $\\sD^{\\operatorname{c}}(R)$ then there are AR\ntriangles in $\\sD^{\\operatorname{c}}(R)$,\n\\[\n  \\Sigma^{-1}(\\operatorname{D}\\!R \\stackrel{\\operatorname{L}}{\\otimes}_R X) \\rightarrow Y \\rightarrow X \\rightarrow\n\\]\nand\n\\[\n  X \n  \\rightarrow Y^{\\prime}\n  \\rightarrow \\Sigma \\operatorname{RHom}_{R^{\\operatorname{o}}}(\\operatorname{D}\\!R,R) \\stackrel{\\operatorname{L}}{\\otimes}_R X\n  \\rightarrow.\n\\]\n\nCombining Equation \\eqref{equ:m} with Theorem \\ref{thm:Gorenstein}(3)\nwhich says $(\\operatorname{D}\\!R)_R \\cong (\\Sigma^d R)_R$ gives\n\\begin{equation}\n\\label{equ:j}\n  \\operatorname{H}(\\tau(-)) \\cong \\operatorname{H}(\\Sigma^{d-1}(-))\n\\end{equation}\nas graded $k$-vector spaces.\n\\end{remark}\n\n\n\n\n\\section{The Auslander-Reiten quiver of a Differential Graded algebra:\n  Local structure}\n\\label{sec:local}\n\n\nThis section considers the AR quiver $\\Gamma$ of the compact derived\ncategory $\\sD^{\\operatorname{c}}(R)$.  When $R$ is Gorenstein with $\\dim_k\n\\operatorname{H}\\!R \\geq 2$, it is proved that each component of $\\Gamma$ is\nisomorphic to ${\\mathbb Z} A_{\\infty}$ as a translation quiver.  The results\nfirst appeared in \\cite{arquiv}; the methods of Karsten Schmidt\n\\cite{Schmidt} have permitted some technical assumptions to be\nremoved.\n\n\\begin{setup}\n\\label{set:local}\nIn this section, $R$ will be Gorenstein with $\\dim_k \\operatorname{H}\\!R \\geq\n2$.\n\nThe category $\\sD^{\\operatorname{c}}(R)$ has AR triangles by Theorem \\ref{thm:R}, and\n${}_{R}k$ is not in $\\sD^{\\operatorname{c}}(R)$ by Theorem \\ref{thm:k}.\n\nThe AR quiver $\\Gamma(\\sD^{\\operatorname{c}}(R))$ will be abbreviated to $\\Gamma$.\n\nThen $\\Gamma$ with the AR translation $\\tau$ is a stable translation\nquiver by Proposition \\ref{pro:stable_translation_quiver}.  By $C$\nwill be denoted a component of the translation quiver $\\Gamma$.\n\\end{setup}\n\n\\begin{lemma}\n\\label{lem:no_loops}\n\\begin{enumerate}\n\n  \\item  No positive power $\\tau^p$ of the AR translation $\\tau$ has a\n    fixed point in $\\Gamma$.\n\n  \\item  $\\Gamma$ has no loops.\n\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(1).  Remark \\ref{rmk:tau} says $\\tau(M) = \\Sigma^{-1}(\\operatorname{D}\\!R\n\\stackrel{\\operatorname{L}}{\\otimes}_R M)$.  Lemma \\ref{lem:inf} implies\n\\[\n  \\inf \\tau(M) = 1 + \\inf \\operatorname{D}\\!R + \\inf M = 1 - d +\\inf M.\n\\]\nSince $d$ is either $0$ or $\\geq 2$, it follows that each positive\npower $\\tau^p(M)$ has $\\inf$ different from $\\inf M$, so no positive\npower is isomorphic to $M$.\n\n(2).  The existence of a loop $[M] \\rightarrow [M]$ would mean the\nexistence of an irreducible morphism $M \\rightarrow M$ in\n$\\sD^{\\operatorname{c}}(R)$.  Such a morphism would be in the radical of the finite\ndimensional algebra $\\operatorname{Hom}_{\\sD^{\\operatorname{c}}(R)}(M,M)$, and hence some power\nwould be zero.  Mimicking the proof of \\cite[lem.\\ VII.2.5]{ARS} now\nshows $\\tau(M) = M$, but this contradicts part (1).\n\\end{proof}\n\nA reference for the graph theoretical terminology of the following\nproposition is \\cite[sec.\\ 4.15]{BensonI}.  A salient fact is that\nwhen $T$ is a directed tree, then the vertices of the repetitive\nquiver ${\\mathbb Z} T$ have the form $(p,t)$ where $p$ is an integer, $t$ is a\nvertex of $T$.  The translation of the stable translation quiver ${\\mathbb Z}\nT$ is determined by $\\tau(p,t) = (p+1,t)$.\n\n\\begin{proposition}\nThere exist a directed tree $T$ and an admissible group of\nautomorphisms $\\Pi$ of ${\\mathbb Z} T$ so that $C \\cong {\\mathbb Z} T\/\\Pi$ as stable\ntranslation quivers.\n\\end{proposition}\n\n\\begin{proof}\nSince $\\tau$ extends to an autoequivalence of $\\sD^{\\operatorname{c}}(R)$ by Remark\n\\ref{rmk:tau}, the AR translation is an automorphism of $\\Gamma$ so\nrestricts to an automorphism of $C$.  By definition, $C$ has no\nmultiple arrows, and by Lemma \\ref{lem:no_loops}(2), it has no\nloops.  Hence the proposition follows from the Riedtmann Structure\nTheorem, see \\cite[thm.\\ 4.15.6]{BensonI}.\n\\end{proof}\n\nTo show that $T = A_{\\infty}$ and that $\\Pi$ acts trivially, the\nfollowing definitions are useful.\n\n\\begin{definition}\n\\label{def:varphi}\nDefine a function on the objects of $\\sD(R)$ by\n\\[\n  \\varphi(M) = \\dim_k \\operatorname{Ext}_R(M,k).\n\\]\nBy abuse of notation, the induced function on the vertices of the AR\nquiver $\\Gamma$ is also denoted by $\\varphi$.\n\nLabel the AR quiver $\\Gamma$ by assigning to the arrow $[M]\n\\stackrel{\\mu}{\\rightarrow} [N]$ the label\n$(\\alpha_{\\mu},\\beta_{\\mu})$, where $\\alpha_{\\mu}$ is the multiplicity\nof $M$ as a direct summand of $Y$ in the AR triangle\n\\[\n  \\tau N \\rightarrow Y \\rightarrow N \\rightarrow\n\\]\nand $\\beta_{\\mu}$ is the multiplicity of $N$ as a direct summand of\n$X$ in the AR triangle\n\\[\n  M \\rightarrow X \\rightarrow \\tau^{-1}M \\rightarrow.\n\\]\n\nThe vertices of ${\\mathbb Z} T$ have the form $(p,t)$ where $p$ is an integer,\n$t$ a vertex of $T$, so each vertex $t$ of $T$ gives a vertex $(0,t)$\nof ${\\mathbb Z} T$ and hence a vertex $\\Pi(0,t)$ of ${\\mathbb Z} T\/\\Pi$, that is, of\n$C$.  Similarly, an arrow $t \\rightarrow t^{\\prime}$ in $T$ gives an\narrow $\\Pi(0,t) \\rightarrow \\Pi(0,t^{\\prime})$ of $C$.  Hence the\nfunction $\\varphi$ and the labelling $(\\alpha,\\beta)$ on\n$\\Gamma$ induce a function and a labelling on $T$.  These will be\ndenoted by $f$ and $(a,b)$.\n\\end{definition}\n\n\\begin{lemma}\n\\label{lem:varphi}\nThe function $\\varphi$ and the labelling $(\\alpha,\\beta)$ have the\nfollowing properties.\n\\begin{enumerate}\n\n  \\item  If $F$ is a minimal semi-free resolution of $M$ with\n    $F^{\\natural} \\cong \\bigoplus_i \\Sigma^i (R^{\\natural})^{(\\beta_i)}$,\n    then $\\varphi(M)$ is equal to the number of direct summands\n    $\\Sigma^i R^{\\natural}$ in $F^{\\natural}$.\n\n  \\item  $\\varphi(\\tau N) = \\varphi(N)$.\n\n  \\item  If $\\tau N \\rightarrow Y \\rightarrow N \\rightarrow$ is an AR\n    triangle in $\\sD^{\\operatorname{c}}(R)$, then $\\varphi(Y) = \\varphi(\\tau N) +\n    \\varphi(N)$.\n\n  \\item If there is an arrow $[M] \\stackrel{\\mu}{\\rightarrow} [N]$ in\n  $\\Gamma$ then there is a corresponding arrow $\\tau[N]\n  \\stackrel{\\nu}{\\rightarrow} [M]$, and $(\\alpha_{\\nu},\\beta_{\\nu}) =\n  (\\beta_{\\mu},\\alpha_{\\mu})$.\n  \n  \\item If there is an arrow $[M] \\stackrel{\\mu}{\\rightarrow} [N]$ in\n  $\\Gamma$ then there is also an arrow $\\tau[M]\n  \\stackrel{\\tau(\\mu)}{\\rightarrow} \\tau[N]$, and\n  $(\\alpha_{\\tau(\\mu)},\\beta_{\\tau(\\mu)}) =\n  (\\alpha_{\\mu},\\beta_{\\mu})$.\n\n  \\item  $\\sum_{\\mu : [M] \\rightarrow [N]}\\alpha_{\\mu}\\varphi(M) =\n    \\varphi(\\tau N) +  \\varphi(N)$, where the sum is over all arrows\n    in $\\Gamma$ with target $[N]$.\n\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(1).  It holds that\n\\[\n  \\varphi(M)\n  = \\dim_k \\operatorname{H}(\\operatorname{RHom}_R(M,k))\n  \\stackrel{\\rm (c)}{=} \\dim_k \\operatorname{H}(\\operatorname{Hom}_R(F,k))\n  \\stackrel{\\rm (d)}{=} \\dim_k \\operatorname{Hom}_{R^{\\natural}}(F^{\\natural},k^{\\natural}),\n\\]\nwhere (c) and (d) are by Lemma \\ref{lem:semi-free}, parts (2) and (5).\nThe right hand side is clearly equal to the number of direct summands\n$\\Sigma^i R^{\\natural}$ in $F^{\\natural}$.\n\n(2).  It holds that\n\\begin{align*}\n  \\varphi(\\tau N)\n  & \\stackrel{\\rm (a)}{=} \\dim_k \\operatorname{H}(\\operatorname{RHom}_R(\\Sigma^{-1}(\\operatorname{D}\\!R \\stackrel{\\operatorname{L}}{\\otimes}_R N),k)) \\\\\n  & = \\dim_k \\operatorname{H}(\\operatorname{RHom}_R(N,\\Sigma \\operatorname{RHom}_R(\\operatorname{D}\\!R,k))) \\\\\n  & \\stackrel{\\rm (b)}{=} \\dim_k \\operatorname{H}(\\operatorname{RHom}_R(N,\\Sigma^{1-d}k)) \\\\\n  & = \\dim_k \\operatorname{H}(\\operatorname{RHom}_R(N,k)) \\\\\n  & = \\varphi(N),\n\\end{align*}\nwhere (a) is by Remark \\ref{rmk:tau} and (b) follows from Theorem\n\\ref{thm:Gorenstein}(3).\n\n(3).  The AR triangle of the lemma induces a long exact sequence\nconsisting of pieces\n\\[\n  \\operatorname{Ext}_R^i(N,k)\n  \\rightarrow \\operatorname{Ext}_R^i(Y,k)\n  \\rightarrow \\operatorname{Ext}_R^i(\\tau N,k),\n\\]\nand the claim will follow if the connecting maps are zero.\n\nIndeed, the AR triangle is also an AR triangle in $\\sD^{\\operatorname{f}}(R)$ by\nProposition \\ref{pro:AR_triangles_preserved}.  A morphism $\\tau N\n\\rightarrow {}_{R}(\\Sigma^i k)$ in $\\sD^{\\operatorname{f}}(R)$ cannot be a split\nmonomorphism since $\\tau N$ is in $\\sD^{\\operatorname{c}}(R)$ while ${}_{R}(\\Sigma^i\nk)$ is not, cf.\\ Setup \\ref{set:local}.  It follows that each such\nmorphism factors through $\\tau N \\rightarrow Y$ whence the composition\n$\\Sigma^{-1}N \\rightarrow \\tau N \\rightarrow \\Sigma^i k$ is zero.\nHence the connecting morphism $\\operatorname{Ext}_R^i(\\tau N,k) \\rightarrow\n\\operatorname{Ext}_R^{i+1}(N,k)$ is zero as desired.\n\n(4).  Let\n\\begin{equation}\n\\label{equ:e}\n  \\tau N \\rightarrow Y \\rightarrow N \\rightarrow\n\\end{equation}\nbe an AR triangle in $\\sD^{\\operatorname{c}}(R)$.  By the definition of the labelling\nof $\\Gamma$, the multiplicity of $M$ as a direct summand of $Y$ is\nequal to both $\\beta_{\\nu}$ and $\\alpha_{\\mu}$, so $\\beta_{\\nu} =\n\\alpha_{\\mu}$.  A similar argument shows $\\alpha_{\\nu} = \\beta_{\\mu}$,\nso $(\\alpha_{\\nu},\\beta_{\\nu}) = (\\beta_{\\mu},\\alpha_{\\mu})$.\n\n(5).  This holds since the AR translation $\\tau$ of $\\sD^{\\operatorname{c}}(R)$ is the\nrestriction of an equivalence of categories by Remark \\ref{rmk:tau}.\n\n(6).  Consider the AR triangle \\eqref{equ:e}.  The object\n$Y$ is a direct sum of copies of the indecomposable objects of\n$\\sD^{\\operatorname{c}}(R)$ which have irreducible morphisms to $N$, and the\nmultiplicity of $M$ as a direct summand of $Y$ is $\\alpha_{\\mu}$ where\n$[M] \\stackrel{\\mu}{\\rightarrow} [N]$ is the arrow in $\\Gamma$.  Hence\n\\[\n  \\sum_{\\mu:[M] \\rightarrow [N]} \\alpha_{\\mu}\\varphi(M) = \\varphi(Y).\n\\]\nNow combine with part (3).\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:zero}\nLet $M \\langle 0 \\rangle, \\ldots, M \\langle 2^p - 1 \\rangle$ be\nindecomposable objects of $\\sD^{\\operatorname{c}}(R)$ with $\\varphi(M \\langle i\n\\rangle) \\leq \\frac{p}{\\dim_k R}$ for each $i$.  If \n\\[\n  M \\langle 2^p - 1 \\rangle \\rightarrow M \\langle 2^p - 2 \\rangle\n  \\rightarrow \\cdots \\rightarrow M \\langle 0 \\rangle\n\\]\nare non-isomorphisms in $\\sD^{\\operatorname{c}}(R)$, then the composition is zero.\n\\end{lemma}\n\n\\begin{proof}\nLet $F \\langle i \\rangle$ be a minimal semi-free resolution of $M\n\\langle i \\rangle$.  Each $F \\langle i \\rangle$ must be\nindecomposable as a DG left-$R$-module, for if $F \\langle i \\rangle$\ndecomposed then it would do so into DG modules $F\n\\langle i_{\\alpha} \\rangle$ with $\\partial(F \\langle i_{\\alpha}\n\\rangle) \\subseteq R^{\\geq 1} \\cdot F \\langle i_{\\alpha} \\rangle$, but\nthis condition forces non-zero cohomology so the decomposition of $F\n\\langle i \\rangle$ as a DG module would induce a non-trivial\ndecomposition of $M \\langle i \\rangle$ in $\\sD^{\\operatorname{c}}(R)$.\n\nThe morphisms in $\\sD^{\\operatorname{c}}(R)$ between the $M \\langle i \\rangle$ are\nrepresented by morphisms\n\\begin{equation}\n\\label{equ:f}\n  F \\langle 2^p - 1 \\rangle \\rightarrow F \\langle 2^p - 2 \\rangle\n  \\rightarrow \\cdots \\rightarrow F \\langle 0 \\rangle\n\\end{equation}\nof DG left-$R$-modules.  These cannot be bijections, since if they\nwere, then the morphisms in $\\sD^{\\operatorname{c}}(R)$ between the $M \\langle i\n\\rangle$ would be isomorphisms.\n\nNow note that if $F \\langle i \\rangle^{\\natural} = \\bigoplus_j\n\\Sigma^j(R^{\\natural})^{(\\beta_j)}$, then the direct sum has\n$\\varphi(M \\langle i \\rangle)$ summands $\\Sigma^j R^{\\natural}$ by\nLemma \\ref{lem:varphi}(1).  Hence\n\\[\n  \\dim_k F \\langle i \\rangle\n  = \\varphi(M \\langle i \\rangle) \\dim_k R\n  \\leq \\frac{p}{\\dim_k R} \\dim_k R\n  = p,\n\\]\nand it is not hard to mimick the proof of \\cite[lem.\\ 4.14.1]{BensonI}\nto see that hence, the composition of the morphisms in Equation\n\\eqref{equ:f} is zero.  This implies that the composition of the\nmorphisms in the lemma is zero.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:non-zero}\nIf $M \\langle 0 \\rangle$ is an indecomposable object of $\\sD^{\\operatorname{c}}(R)$\nand $q \\geq 0$ is an integer, then there exist indecomposable objects\nand irreducible morphisms in $\\sD^{\\operatorname{c}}(R)$,\n\\[\n  M \\langle q \\rangle \\rightarrow M \\langle q-1 \\rangle\n  \\rightarrow \\cdots \\rightarrow M \\langle 0 \\rangle,\n\\]\nwith non-zero composition.\n\\end{lemma}\n\n\\begin{proof}\nLet me prove a stronger statement which implies the lemma: If $M\n\\langle 0 \\rangle$ is an indecomposable object of $\\sD^{\\operatorname{c}}(R)$ and $q\n\\geq 0$ is an integer, then there exists\n\\[\n  {}_{R}(\\Sigma^i k)\n  \\stackrel{\\kappa_q}{\\rightarrow} M \\langle q \\rangle\n  \\stackrel{\\mu_q}{\\rightarrow} M \\langle q-1 \\rangle\n  \\stackrel{\\mu_{q-1}}{\\rightarrow} \\cdots\n  \\stackrel{\\mu_1}{\\rightarrow} M \\langle 0 \\rangle\n\\]\nwhere the $M \\langle i \\rangle$ are indecomposable objects of\n$\\sD^{\\operatorname{c}}(R)$ and the $\\mu_i$ are irreducible morphisms in $\\sD^{\\operatorname{c}}(R)$,\nsuch that $\\mu_1 \\circ \\cdots \\circ \\mu_q \\circ \\kappa_q \\neq 0$.\n\nUsing induction on $q$, first let $q = 0$.  Let $F$ be a minimal\nsemi-free resolution of the dual $\\operatorname{D}\\!M \\langle 0 \\rangle$.  Then\n\\begin{align*}\n  \\operatorname{H}(\\operatorname{RHom}_R(k,M \\langle 0 \\rangle))\n  & \\cong \\operatorname{H}(\\operatorname{RHom}_{R^{\\operatorname{o}}}(\\operatorname{D}\\!M \\langle 0 \\rangle,k)) \\\\\n  & \\stackrel{\\rm (a)}{\\cong} \\operatorname{H}(\\operatorname{Hom}_{R^{\\operatorname{o}}}(F,k)) \\\\\n  & \\stackrel{\\rm (b)}{\\cong} \\operatorname{Hom}_{(R^{\\operatorname{o}})^{\\natural}}(F^{\\natural},k^{\\natural}) \\\\\n  & \\stackrel{\\rm (c)}{\\not\\cong} 0.\n\\end{align*}\nHere (a) and (b) are by Lemma \\ref{lem:semi-free}, parts (2) and (5).\n(c) is because $M \\langle 0 \\rangle$ is indecomposable hence has\nnon-zero cohomology; this implies that $\\operatorname{D}\\!M \\langle 0 \\rangle$\nhas non-zero cohomology, and then $F$ is non-trivial semi-free whence\n$F^{\\natural}$ is a non-trivial graded free module.\n\nIt follows from the displayed formula that there is a non-zero\nmorphism\n\\[\n  {}_{R}(\\Sigma^i k) \\stackrel{\\kappa_0}{\\rightarrow} M \\langle 0 \\rangle\n\\]\nfor some $i$.\n\nNow let $q \\geq 1$ and suppose that\n\\[\n  {}_{R}(\\Sigma^i k)\n  \\stackrel{\\kappa_{q-1}}{\\rightarrow} M \\langle q-1 \\rangle\n  \\stackrel{\\mu_{q-1}}{\\rightarrow} M \\langle q-2 \\rangle\n  \\stackrel{\\mu_{q-2}}{\\rightarrow} \\cdots\n  \\stackrel{\\mu_1}{\\rightarrow} M \\langle 0 \\rangle\n\\]\nhas already been found with the desired properties.  Let $\\tau M\n\\langle q-1 \\rangle \\rightarrow X \\langle q \\rangle\n\\stackrel{\\mu_{q}^{\\prime}}{\\rightarrow} M \\langle q-1 \\rangle\n\\rightarrow$ be an AR triangle in $\\sD^{\\operatorname{c}}(R)$.  By Proposition\n\\ref{pro:AR_triangles_preserved} it is also an AR triangle in\n$\\sD^{\\operatorname{f}}(R)$.  Since ${}_{R}k$ is not in $\\sD^{\\operatorname{c}}(R)$, see Setup\n\\ref{set:local}, it is clear that $\\kappa_{q-1}$ is not a split\nepimorphism, so it factors through $\\mu_{q}^{\\prime}$.  Now I can get\nthe situation claimed in the lemma by letting $M \\langle q \\rangle$ be\na suitable indecomposable summand of $X \\langle q \\rangle$ and $\\mu_q$\nthe restriction of $\\mu^{\\prime}_q$ to $M \\langle q \\rangle$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:varphi_unbounded}\nThe function $\\varphi$ is unbounded on $C$. \n\\end{lemma}\n\n\\begin{proof}\nIf $\\varphi$ were bounded on $C$ then Lemma \\ref{lem:zero} would apply\nto sufficiently long sequences of morphisms between indecomposable\nobjects with vertices in $C$, but this would make impossible the\nsituation established in Lemma \\ref{lem:non-zero}.\n\\end{proof}\n\nRecall that the Cartan matrix $c$ of the labelled directed tree $T$ is\na matrix with rows and columns indexed by the vertices of $T$.  If $s$\nand $t$ are vertices, then\n\\[\n  c_{st} =\n  \\left\\{\n    \\begin{array}{cl}\n      2 & \\mbox{if $s=t$}, \\\\\n      -a_{\\mu} & \\mbox{if there is an arrow $s \\stackrel{\\mu}{\\rightarrow} t$}, \\\\\n      -b_{\\nu} & \\mbox{if there is an arrow $t \\stackrel{\\nu}{\\rightarrow} s$}, \\\\\n      0 & \\mbox{if $s \\neq t$ and $s$ and $t$ are not connected by an arrow};\n    \\end{array}\n  \\right.\n\\]\ncp.\\ \\cite[sec.\\ 4.5]{BensonI}.\nThe function $f$ on the vertices of $T$ is called additive if it\nsatisfies $\\sum_s c_{st}f(s) = 0$ for each $t$, that is,\n\\begin{equation}\n\\label{equ:g}\n  2f(t) - \\sum_{\\mu : s \\rightarrow t}a_{\\mu}f(s)\n        - \\sum_{\\nu : t \\rightarrow u}b_{\\nu}f(u) = 0\n\\end{equation}\nfor each $t$, where the sums are over all arrows in $T$ into $t$ and\nout of $t$.  Indeed:\n\n\\begin{proposition}\n\\label{pro:f}\nThe function $f$ is additive and unbounded on $T$.\n\\end{proposition}\n\n\\begin{proof}\nUsing Definition \\ref{def:varphi}, the left hand side of Equation\n\\eqref{equ:g} can be rewritten\n\\[\n  2\\varphi(\\Pi(0,t))\n  - \\sum_{\\mu : s \\rightarrow t} \\alpha_{\\Pi(0,s) \\rightarrow \\Pi(0,t)} \\varphi(\\Pi(0,s))\n  - \\sum_{\\nu : t \\rightarrow u} \\beta_{\\Pi(0,t) \\rightarrow \\Pi(0,u)} \\varphi(\\Pi(0,u)).\n\\]\nThe translation of ${\\mathbb Z} T\/\\Pi$ is given by $\\tau(\\Pi(p,t)) =\n\\Pi(p+1,t)$.  To each arrow $\\Pi(0,t) \\rightarrow \\Pi(0,u)$\ncorresponds an arrow $\\tau(\\Pi(0,u)) \\rightarrow \\Pi(0,t)$, that is,\n$\\Pi(1,u) \\rightarrow \\Pi(0,t)$.  Lemma \\ref{lem:varphi}(4) gives\n$\\beta_{\\Pi(0,t) \\rightarrow \\Pi(0,u)} = \\alpha_{\\Pi(1,u) \\rightarrow\n\\Pi(0,t)}$.  Lemma \\ref{lem:varphi}(2) gives $\\varphi(\\Pi(0,u)) =\n\\varphi(\\tau(\\Pi(0,u))) = \\varphi(\\Pi(1,u))$, and also implies that\n$2\\varphi(\\Pi(0,t)) = \\varphi(\\Pi(0,t)) + \\varphi(\\tau(\\Pi(0,t)))$.\n\nSubstituting all this into the previous expression gives\n\\begin{align*}\n  \\lefteqn{\\varphi(\\Pi(0,t)) + \\varphi(\\tau(\\Pi(0,t)))} & \\\\\n  & \\hspace{10ex} - \\sum_{\\mu : s \\rightarrow t} \\alpha_{\\Pi(0,s) \\rightarrow \\Pi(0,t)} \\varphi(\\Pi(0,s))\n  - \\sum_{\\nu : t \\rightarrow u} \\alpha_{\\Pi(1,u) \\rightarrow \\Pi(0,t)} \\varphi(\\Pi(1,u)).\n\\end{align*}\nRecall that the sums are over all the arrows in $T$ into $t$ and out\nof $t$.  From the construction of the repetitive quiver ${\\mathbb Z} T$, this\nmeans that between them, the sums can be viewed as being over all the\narrows into $(0,t)$ in ${\\mathbb Z} T$.  However, the projection ${\\mathbb Z} T\n\\rightarrow {\\mathbb Z} T\/\\Pi$ is a covering so induces a bijection between\nthe arrows in ${\\mathbb Z} T$ into $(0,t)$ and the arrows in ${\\mathbb Z} T\/\\Pi$ into\n$\\Pi(0,t)$.  So in fact, the previous expression can be rewritten\n\\[\n  \\varphi(\\Pi(0,t)) + \\varphi(\\tau(\\Pi(0,t)))\n  - \\sum_{m \\rightarrow \\Pi(0,t)} \\alpha_{m \\rightarrow \\Pi(0,t)} \\varphi(m)\n\\]\nwhere the sum is over all arrows in ${\\mathbb Z} T\/\\Pi$ into $\\Pi(0,t)$.  But\nidentifying ${\\mathbb Z} T\/\\Pi$ and $C$, the displayed expression is zero by\nLemma \\ref{lem:varphi}(6), so $f$ is additive.\n\nSince $f(t) = \\varphi(\\Pi(0,t))$ by Definition \\ref{def:varphi} and\n$\\varphi(\\Pi(p,t)) = \\varphi(\\tau^p\\Pi(0,t)) = \\varphi(\\Pi(0,t))$ by\nLemma \\ref{lem:varphi}(2), if $f$ were bounded on $T$ then $\\varphi$\nwould be bounded on $C$.  But this is false by Lemma\n\\ref{lem:varphi_unbounded}.\n\\end{proof}\n\nRecall that the graph $A_{\\infty}$ is\n\\[\n  \\def\\scriptstyle{\\scriptstyle}\n  \\vcenter{\n  \\xymatrix @!0 @+0.25pc {\n    1 \\ar@{-}[rr] & & 2 \\ar@{-}[rr] & & 3 \\ar@{-}[rr] & & 4 \\ar@{-}[rr] & & 5 \\ar@{-}[rr]&&{\\textstyle \\cdots}\\\\\n                         }\n          },\n\\]\nwhere a convenient numbering of the vertices has been chosen.  A\nquiver of type $A_{\\infty}$ is an orientation of this graph.  The\nrepetitive quiver ${\\mathbb Z} A_{\\infty}$ does not depend on the orientation;\nwith a standard numbering of the vertices it is\n\\[\n  \\def\\scriptstyle{\\scriptstyle}\n  \\vcenter{\n  \\xymatrix @!0 @+0.5pc {\n    & & & \\vdots & & & & \\vdots & & & \\\\\n    & *{(3,5)} \\ar[dr] & & *{(2,5)} \\ar[dr] & & *{(1,5)} \\ar[dr] & & *{(0,5)} \\ar[dr] & & *{(-1,5)} & \\\\\n    & & *{(2,4)} \\ar[dr] \\ar[ur] & & *{(1,4)} \\ar[dr] \\ar[ur] & & *{(0,4)} \\ar[dr] \\ar[ur] & & *{(-1,4)} \\ar[dr] \\ar[ur] & & \\\\\n    {\\textstyle \\cdots} & *{(2,3)} \\ar[dr] \\ar[ur] & *{} & *{(1,3)} \\ar[dr] \\ar[ur] & *{} & *{(0,3)} \\ar[dr] \\ar[ur] & *{} & *{(-1,3)} \\ar[dr] \\ar[ur] & *{} & *{(-2,3)} & {\\textstyle \\cdots}.\\\\\n    & & *{(1,2)} \\ar[dr] \\ar[ur] & & *{(0,2)} \\ar[dr] \\ar[ur] & & *{(-1,2)} \\ar[dr] \\ar[ur] & & *{(-2,2)} \\ar[dr] \\ar[ur] & & \\\\\n    & *{(1,1)} \\ar[ur] & & *{(0,1)} \\ar[ur] & & *{(-1,1)} \\ar[ur] & & *{(-2,1)} \\ar[ur] & & *{(-3,1)} & \\\\\n                      }\n          }\n\\]\nThe translation acts by $\\tau(p,t) = (p+1,t)$.\n\n\\begin{theorem}\n\\label{thm:quiver_local_structure}\n\\begin{enumerate}\n  \\item  The component $C$ of the AR quiver $\\Gamma$ of $\\sD^{\\operatorname{c}}(R)$ is\n    isomorphic to ${\\mathbb Z} A_{\\infty}$ as a stable translation quiver.\n\n  \\item  Each label $(\\alpha_{\\mu},\\beta_{\\mu})$ on $\\Gamma$ is equal\n    to $(1,1)$. \n\n  \\item  If the function $\\varphi$ has value $\\varphi_1$ on the edge of\n    $C \\cong {\\mathbb Z} A_{\\infty}$, then it has value $n\\varphi_1$ on the\n    $n$'th horizontal row of vertices in $C$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nBy Proposition \\ref{pro:f}, there is an additive unbounded function\n$f$ on the labelled tree $T$.  Hence $T$ is of type $A_{\\infty}$ with\nall labels equal to $(1,1)$ by \\cite[thm.\\ 4.5.8(iv)]{BensonI}.  This\nproves (2), and it also means that to prove (1), it is sufficient to\nshow that $\\Pi$ acts trivially on ${\\mathbb Z} A_{\\infty}$.\n\nBut if it did not, then there would exist a vertex $m$ on the edge of\n${\\mathbb Z} A_{\\infty}$ and a $g$ in $\\Pi$ such that $gm \\neq m$.  The vertex\n$gm$ would again be on the edge, and so it would have the form $\\tau^p\nm$ for some $p \\neq 0$.  But then $m$ and $\\tau^p m$ would get\nidentified in ${\\mathbb Z} A_{\\infty}\/\\Pi$, and hence $\\Pi m$ would be a fixed\npoint in ${\\mathbb Z} A_{\\infty}\/\\Pi$ of $\\tau^p$, that is, a fixed point in\n$C$ of $\\tau^p$.  But this is impossible by Lemma\n\\ref{lem:no_loops}(1).\n\nFinally, it is a standard consequence of additivity that if the\nfunction $f$ has value $f(1) = f_1$ at the first vertex of\n$A_{\\infty}$, then it has value $f(n) = nf_1$ at the $n$th vertex.\nSince $\\varphi(\\Pi(p,n)) = \\varphi(\\tau^p(\\Pi(0,n))) =\n\\varphi(\\Pi(0,n)) = f(n)$, the claim (3) on $\\varphi$ follows.\n\\end{proof}\n\n\n\n\n\\section{Report on work by Karsten Schmidt}\n\\label{sec:global}\n\n\nIn this section, the study of the AR quiver $\\Gamma$ of $\\sD^{\\operatorname{c}}(R)$ is\ncontinued, and some aspects of the global structure are revealed.  If\n$\\dim_k \\operatorname{H}\\!R = 2$ then $\\Gamma$ has precisely $d-1$\ncomponents.  On the other hand, for Gorenstein algebras with $\\dim_k\n\\operatorname{H}\\!R \\geq 3$, there are infinitely many components.  Often,\nthese even form families which are indexed by projective manifolds,\nand these manifolds can be of arbitrarily high dimension.\n\nWith the exception of Theorem \\ref{thm:spheres} which is essentially\nin \\cite{artop}, the results of this section are due to Karsten\nSchmidt; see \\cite[thm.\\ 4.1]{Schmidt}.\n\nOnly a sketch is given of the proof of the next theorem; for more\ninformation, see \\cite[sec.\\ 8]{artop}.\n\n\\begin{theorem}\n\\label{thm:spheres}\nIf $\\dim_k \\operatorname{H}\\!R = 2$ then $\\sD^{\\operatorname{c}}(R)$ has AR triangles, and\nthe AR quiver of $\\sD^{\\operatorname{c}}(R)$ has $d-1$ components, each isomorphic to\n${\\mathbb Z} A_{\\infty}$.\n\\end{theorem}\n\n\\begin{proof}\nThe cohomology of $R$ in low degrees is $\\operatorname{H}^0\\!R = k$ and\n$\\operatorname{H}^1\\!R = 0$.  Since $\\dim_k \\operatorname{H}\\!R = 2$, it follows\nthat the only other non-zero cohomology is $\\operatorname{H}^d\\!R = k$, and\nit is easy to check that $R$ therefore satisfies the conditions of\nTheorem \\ref{thm:Gorenstein}(2) so $R$ is Gorenstein.  Theorem\n\\ref{thm:R} says that $\\sD^{\\operatorname{c}}(R)$ has AR triangles, and Theorem\n\\ref{thm:quiver_local_structure}(1) says that each component of the AR\nquiver of $\\sD^{\\operatorname{c}}(R)$ is isomorphic to ${\\mathbb Z} A_{\\infty}$.\n\nReplacing $R$ with a quasi-isomorphic truncation, it can be supposed\nthat $R^{>d} = 0$, see Lemma \\ref{lem:truncations}(3).  Pick a cycle\n$x$ in $R^d$ with non-zero cohomology class.  The graded algebra\n$k[X]\/(X^2)$ with $X$ in cohomological degree $d$ can be viewed as a\nDG algebra with zero differential, and the map $k[X]\/(X^2) \\rightarrow\nR$ sending $X$ to $x$ is a quasi-isomorphism, so $R$ can be replaced\nwith $k[X]\/(X^2)$.\n\nNow consider the algebra $S = k[Y]$ with $Y$ in cohomological degree\n$-d + 1$, viewed as a DG algebra with zero differential.  The DG\nmodule $k$ can be viewed as a DG right-$R$-right-$S$-module in an\nobvious way, and it induces adjoint functors\n\\[\n  \\xymatrix{\n    \\sD(S^{\\operatorname{o}}) \\ar@<-1ex>[rr]_{\\operatorname{RHom}_{S^{\\operatorname{o}}}(k,-)}\n    & & \\sD(R). \\ar@<-1ex>[ll]_{k \\stackrel{\\operatorname{L}}{\\otimes}_R -}\n           }\n\\]\nThe upper functor clearly sends ${}_{R}R$ to $k_S$, and by computing\nwith a semi-free resolution it can be verified that the lower functor\nsends $k_S$ to ${}_{R}R$.  Hence the functors restrict to\nquasi-inverse equivalences on the subcategories of objects which are\nfinitely built, respectively, from $k_S$ and ${}_{R}R$.  These\nsubcategories are precisely $\\sD^{\\operatorname{f}}(S^{\\operatorname{o}})$ and $\\sD^{\\operatorname{c}}(R)$.\n\nSo it is enough to show that the AR quiver of $\\sD^{\\operatorname{f}}(S^{\\operatorname{o}})$ has\n$d-1$ components.  However, $S$ is $k[Y]$ equipped with zero\ndifferential, so $\\operatorname{H}\\!S$ is just $k[Y]$ viewed as a graded\nalgebra.  This polynomial algebra in one variable has global dimension\n$1$, and this makes it possible to prove that if $M$ is a DG\nright-$S$-module, then $M$ is quasi-isomorphic to $\\operatorname{H}\\!M$\nequipped with zero differential.\n\nThis reduces the classification of objects of $\\sD^{\\operatorname{f}}(S^{\\operatorname{o}})$ to\nthe classification of graded right-$\\operatorname{H}\\!S$-modules.  However,\nusing again that $\\operatorname{H}\\!S = k[Y]$ is a polynomial algebra in one\nvariable, one shows that its indecomposable finite dimensional graded\nright-modules are precisely\n\\[\n  \\Sigma^j k[Y]\/(Y^{m+1})\n\\]\nfor $j$ in ${\\mathbb Z}$ and $m \\geq 0$.  Viewing these as DG\nright-$S$-modules with zero differential gives the indecomposable\nobjects of $\\sD^{\\operatorname{f}}(S^{\\operatorname{o}})$, and knowing the indecomposable objects,\nit is an exercise in AR theory to compute the AR triangles, find the\nAR quiver, and verify that it has $d-1$ components.\n\\end{proof}\n\n\\begin{setup}\nIn the rest of this section, the setup of Section \\ref{sec:local} will\nbe kept: $R$ is Gorenstein with $\\dim_k \\operatorname{H}\\!R \\geq 2$.\n\nThe category $\\sD^{\\operatorname{c}}(R)$ has AR triangles by Theorem \\ref{thm:R}, and\n${}_{R}k$ is not in $\\sD^{\\operatorname{c}}(R)$ by Theorem \\ref{thm:k}.\n\nThe AR quiver $\\Gamma(\\sD^{\\operatorname{c}}(R))$ will abbreviated to $\\Gamma$.\n\\end{setup}\n\nSince $\\operatorname{H}^0\\!R \\cong k$ and $\\operatorname{H}^1\\!R = 0$, by Theorem\n\\ref{thm:Gorenstein}(2) it must be the case that $\\operatorname{H}^{d-1}\\!R =\n0$ and $\\operatorname{H}^d\\!R \\cong k$.  By definition, $d$ is the highest\ndegree in which $R$ has non-zero cohomology; suppose that $e \\not\\in \\{\n0,d \\}$ is another degree with $\\operatorname{H}^e\\!R \\neq 0$ and observe\nthat then\n\\[\n  2 \\leq e \\leq d-2\n\\]\nand $d \\geq 4$.\n\nLet $X$ be a minimal semi-free DG left-$R$-module whose semi-free\nfiltration contains only finitely many copies of (de)suspensions of\n$R$.  In particular, Lemma \\ref{lem:semi-free}(4) says that $X$ is in\n$\\sD^{\\operatorname{c}}(R)$; suppose that it is indecomposable in that category.  Let\n$i \\geq 2$ and consider the following cases.\n\n\\begin{description}\n\n  \\item[Case (1).] Suppose that\n\\[\n  \\mbox{$\\inf X = 0 \\;\\;$ and $\\;\\; \\sup X = i$.}\n\\]\nA non-zero cohomology class in $\\operatorname{H}^i\\!X$ permits a\nnon-zero morphism $\\Sigma^{-i}R \\stackrel{g}{\\rightarrow} X$;\ndenoting the mapping cone by $X(1)$, there is a distinguished triangle\n\\begin{equation}\n\\label{equ:1}\n  \\Sigma^{-i}R \\stackrel{g}{\\rightarrow} X \\rightarrow X(1) \\rightarrow.\n\\end{equation}\n\n  \\item[Case (2).]  Suppose that\n\\[\n  \\mbox{$\\inf X = 0, \\;\\; \\sup X = i, \\;\\;$\n        and $\\operatorname{H}^{i-d+e}X \\neq 0$.}\n\\]\nA non-zero cohomology class in $\\operatorname{H}^{i-d+e}\\!X$ permits a\nnon-zero morphism $\\Sigma^{-i+d-e}R \\stackrel{h}{\\rightarrow} X$;\ndenoting the mapping cone by $X(2)$, there is a distinguished triangle\n\\begin{equation}\n\\label{equ:2}\n  \\Sigma^{-i+d-e}R \\stackrel{h}{\\rightarrow} X \\rightarrow X(2) \\rightarrow.\n\\end{equation}\n\n  \\item[Case (2${}_{\\alpha}$).]  In Case (2), suppose moreover\n  that $\\operatorname{H}^i\\!X \\cong k$ and that scalar multiplication induces\n  a non-degenerate bilinear form\n\\begin{equation}\n\\label{equ:b}\n  \\operatorname{H}^{d-e}(R) \\times \\operatorname{H}^{i-d+e}(X)\n  \\rightarrow \\operatorname{H}^i(X) \\cong k.\n\\end{equation}\n\nThe morphism $\\Sigma^{-i+d-e}R \\stackrel{h}{\\rightarrow} X$\ncorresponds to an element $\\alpha$ in $\\operatorname{H}^{i-d+e}\\!X$; denote\n$h$ by $h_{\\alpha}$ and $X(2)$ by $X(2_{\\alpha})$.\n\n\\end{description}\n\nIt follows from the mapping cone construction that $X(1)$, $X(2)$, and\n$X(2_{\\alpha})$ are again minimal semi-free DG left-$R$-modules whose\nsemi-free filtrations contain only fi\\-ni\\-te\\-ly many copies of\n(de)suspensions of $R$.\n\n\\begin{lemma}\n\\label{lem:C}\n\\begin{enumerate}\n\n  \\item In Case (1) of the above construction, the DG module $X(1)$ is\n  indecomposable in $\\sD^{\\operatorname{c}}(R)$.  It has\n\\[\n  \\inf X(1) = 0, \\;\\; \\sup X(1) = i+d-1\n\\]\nand\n\\[\n  \\operatorname{H}^{i+e-1}(X(1)) \\cong \\operatorname{H}^e(R) \\neq 0, \\;\\;\n  \\operatorname{H}^{i+d-1}(X(1)) \\cong \\operatorname{H}^d(R) \\cong k.\n\\]\nIt satisfies $\\operatorname{amp}(X(1)) = \\operatorname{amp}(X) + d - 1$ and $\\varphi(X(1)) =\n\\varphi(X) + 1$.  Moreover, if the construction is applied to $X$ and\n$X^{\\prime}$ then $X(1) \\cong X^{\\prime}(1)$ implies $X \\cong\nX^{\\prime}$ in $\\sD^{\\operatorname{c}}(R)$.  Finally, scalar multiplication induces a\nnon-degenerate bilinear form\n\\[\n  \\operatorname{H}^{d-e}(R) \\times \\operatorname{H}^{i+e-1}(X(1))\n  \\rightarrow \\operatorname{H}^{i+d-1}(X(1)) \\cong k.\n\\]\n\n  \\item In Case (2), the DG module $X(2)$ is indecomposable in\n  $\\sD^{\\operatorname{c}}(R)$.  It has\n\\[\n  \\mbox{$\\inf X(2) = 0 \\;\\;$ and $\\;\\; \\sup X(2) = i+e-1$}.\n\\]\nIt satisfies $\\operatorname{amp}(X(2)) = \\operatorname{amp}(X) + e - 1$ and $\\varphi(X(2)) =\n\\varphi(X) + 1$.  Moreover, if the construction is applied to $X$ and\n$X^{\\prime}$ then $X(2) \\cong X^{\\prime}(2)$ implies $X \\cong\nX^{\\prime}$ in $\\sD^{\\operatorname{c}}(R)$.\n\n  \\item In Case (2${}_{\\alpha}$), if $\\alpha$ and $\\alpha^{\\prime}$ are\n  elements of $\\operatorname{H}^{i-d+e}\\!X$ then\n\\[\n  \\mbox{$X(2_{\\alpha}) \\cong X(2_{\\alpha^{\\prime}})$ in $\\sD^{\\operatorname{c}}(R)$\n    \\; $\\Leftrightarrow$ \\;\n    $\\alpha = \\kappa\\alpha^{\\prime}$ for a $\\kappa$ in $k$.} \n\\]\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(1).  Indecomposability will follow from \\cite[lem.\\ 6.5]{HKR} if\nI can show in $\\sD^{\\operatorname{c}}(R)$ that $g$ is non-zero (clear), non-invertible\n(clear since $\\inf \\Sigma^{-i}R = i \\geq 2$ but $\\inf X = 0$), and\nthat $\\operatorname{Hom}_{\\sD^{\\operatorname{c}}(R)}(X,\\Sigma\\Sigma^{-i}R) = 0$.  However,\n\\begin{align*}\n  \\operatorname{Hom}_{\\sD^{\\operatorname{c}}(R)}(X,\\Sigma\\Sigma^{-i}R)\n  & \\cong \\operatorname{Hom}_{\\sD^{\\operatorname{c}}(R^{\\operatorname{o}})}(\\operatorname{D}\\!\\Sigma\\Sigma^{-i}R,\\operatorname{D}\\!X) \\\\\n  & \\cong \\operatorname{Hom}_{\\sD^{\\operatorname{c}}(R^{\\operatorname{o}})}(\\Sigma^{i-1}\\operatorname{D}\\!R,\\operatorname{D}\\!X) \\\\\n  & \\stackrel{\\rm (a)}{\\cong} \\operatorname{Hom}_{\\sD^{\\operatorname{c}}(R^{\\operatorname{o}})}(\\Sigma^{i-1+d}R,\\operatorname{D}\\!X) \\\\\n  & \\cong \\operatorname{H}^{-i+1-d}(\\operatorname{D}\\!X) \\\\\n  & \\cong \\operatorname{D}\\!\\operatorname{H}^{i-1+d}(X) \\\\\n  & \\stackrel{\\rm (b)}{=} 0\n\\end{align*}\nwhere (a) is by Theorem \\ref{thm:Gorenstein}(3) and (b) is because\n$\\sup X = i$.\n\nThe statements $\\inf X(1) = 0$, $\\sup X(1) = i+d-1$,\n$\\operatorname{H}^{i+e-1}(X(1)) \\cong \\operatorname{H}^e(R) \\neq 0$, and\n$\\operatorname{H}^{i+d-1}(X(1)) \\cong \\operatorname{H}^d(R) \\cong k$ follow from the\nlong exact cohomology sequence of the distinguished triangle\n\\eqref{equ:1}.  The statement about the amplitude is a consequence,\nand $\\varphi(X(1)) = \\varphi(X) + 1$ because $X(1)$ is minimal\nsemi-free with one more copy of a desuspension of $R$ in its\nsemi-free filtration than $X$; cf. Lemma \\ref{lem:varphi}(1).\n\nTo get the statement on isomorphisms, first observe that by a\ncomputation like the one above,\n\\[\n  \\operatorname{Hom}_{\\sD^{\\operatorname{c}}(R)}(X(1),\\Sigma\\Sigma^{-i}R)\n  \\cong \\operatorname{D}\\!\\operatorname{H}^{i+d-1}(X(1))\n  \\cong \\operatorname{D}(k)\n  \\cong k.\n\\]\nNow suppose that there is an isomorphism $X(1)\n\\stackrel{\\sim}{\\rightarrow} X^{\\prime}(1)$ in $\\sD^{\\operatorname{c}}(R)$.  This\ngives a diagram between the distinguished triangles defining $X(1)$\nand $X^{\\prime}(1)$,\n\\[\n  \\xymatrix{\n  \\Sigma^{-i}R \\ar[r] & X \\ar[r] & X(1) \\ar[r] \\ar[d] & \\Sigma^{-i+1}R \\\\\n  \\Sigma^{-i}R \\ar[r] & X^{\\prime} \\ar[r] & X^{\\prime}(1) \\ar[r] & \\Sigma^{-i+1}R \\lefteqn{.}\n           }\n\\]\nThe last morphism in the upper distinguished triangle is non-zero, for\notherwise the triangle would be split contradicting that $X$ is\nindecomposable.  Since $\\operatorname{Hom}_{\\sD^{\\operatorname{c}}(R)}(X(1),\\Sigma\\Sigma^{-i}R)$ is\none-dimensional, there exists a morphism $\\Sigma^{-i+1}R \\rightarrow\n\\Sigma^{-i+1}R$ to give a commutative square.  Adding this morphism\nand its desuspension to the diagram gives\n\\[\n  \\xymatrix{\n  \\Sigma^{-i}R \\ar[r] \\ar[d] & X \\ar[r] & X(1) \\ar[r] \\ar[d] & \\Sigma^{-i+1}R \\ar[d]\\\\\n  \\Sigma^{-i}R \\ar[r] & X^{\\prime} \\ar[r] & X^{\\prime}(1) \\ar[r] & \\Sigma^{-i+1}R \\lefteqn{,}\n           }\n\\]\nand the two new vertical arrows are also isomorphisms since they are\nnon-zero and since $\\operatorname{Hom}_{\\sD^{\\operatorname{c}}(R)}(R,R) \\cong k$.  By the axioms of\ntriangulated categories, there is a vertical morphism $X \\rightarrow\nX^{\\prime}$ which completes to a commutative diagram, and this\nmorphism is an isomorphism by the triangulated five lemma.\n\nFinally, to get the non-degenerate bilinear form, observe that $R$ is\nGorenstein so by Theorem \\ref{thm:Gorenstein}(2) scalar multiplication\ngives a non-degenerate bilinear form\n\\[\n  \\operatorname{H}^{d-e}(R) \\times \\operatorname{H}^{i+e-1}(\\Sigma^{-i+1}R)\n  \\rightarrow \\operatorname{H}^{i+d-1}(\\Sigma^{-i+1}R) \\cong k.\n\\]\nBut $X(1)$ is a mapping cone which in degrees $\\geq i+1$ is equal to\n$\\Sigma^{-i+1}R$, so this gives a non-degenerate bilinear form\n\\[\n  \\operatorname{H}^{d-e}(R) \\times \\operatorname{H}^{i+e-1}(X(1))\n  \\rightarrow \\operatorname{H}^{i+d-1}(X(1)) \\cong k\n\\]\nas claimed.\n\n(2)  follows by similar arguments.\n\n(3).  $\\Leftarrow$ is elementary.  $\\Rightarrow$:  Given the isomorphism\n$X(2_{\\alpha}) \\rightarrow X(2_{\\alpha^{\\prime}})$, the method applied\nin the proof of (1) produces a diagram between the distinguished\ntriangles defining $X(2_{\\alpha})$ and $X(2_{\\alpha^{\\prime}})$,\n\\[\n  \\xymatrix{\n  \\Sigma^{-i+d-e}R \\ar[r]^-{h_{\\alpha}} \\ar[d] & X \\ar[r] \\ar[d]^{\\gamma} & X(2_{\\alpha}) \\ar[r] \\ar[d] & \\Sigma^{-i+d-e+1}R \\ar[d]\\\\\n  \\Sigma^{-i+d-e}R \\ar[r]_-{h_{\\alpha^{\\prime}}} & X \\ar[r] & X(2_{\\alpha^{\\prime}}) \\ar[r] & \\Sigma^{-i+d-e+1}R \\lefteqn{,}\n           }\n\\]\nwhere the vertical maps are isomorphisms.  Commutativity of the first\nsquare implies $(\\operatorname{H}^{i-d+e}(\\gamma))(\\alpha) = \\alpha^{\\prime}$.\n\nConsider $x$ in $\\operatorname{H}^{d-e}\\!R$.  Then\n\\[\n  x\\alpha^{\\prime} = 0\n  \\Leftrightarrow x(\\operatorname{H}^{i-d+e}(\\gamma))(\\alpha) = 0\n  \\Leftrightarrow (\\operatorname{H}^{i-d+e}(\\gamma))(x\\alpha) = 0\n  \\Leftrightarrow x\\alpha = 0,\n\\]\nthe last $\\Leftrightarrow$ because $\\gamma$ is an isomorphism in\n$\\sD^{\\operatorname{c}}(R)$ whence $\\operatorname{H}^{i-d+e}(\\gamma)$ is bijective.  Seeing\nthat the bilinear form \\eqref{equ:b} is non-degenerate, this means\nthat $\\alpha = \\kappa\\alpha^{\\prime}$ for a $\\kappa$ in $k$.\n\\end{proof}\n\nObserve that it makes sense to insert $X(1)$ into either of Cases (1),\n(2), and (2${}_{\\alpha}$).  Likewise, it makes sense to insert $X(2)$\nand $X(2_{\\alpha})$ into Case (1).  Iterating Cases (1) and (2), the\nfollowing tree can be constructed.\n\\begin{equation}\n\\label{equ:tree}\n  \\vcenter{\n    \\xymatrix @C+1pc @R-2pc {\n      & & & X(1,1,1) & \\\\\n      & & X(1,1) \\ar@{.}[ur] \\ar@{.}[dr] & & \\ldots \\\\\n      & & & X(1,1,2) & \\\\\n      & X(1) \\ar@{.}[uur] \\ar@{.}[ddr] & & & \\\\\n      & & & X(1,2,1) & \\\\\n      & & X(1,2) \\ar@{.}[ur] \\ar@{.}[dr] & & \\\\\n      & & & {*} & \\\\\n      X \\ar@{.}[uuuur] \\ar@{.}[ddddr] & & & & \\dots \\\\\n      & & & X(2,1,1) & \\\\\n      & & X(2,1) \\ar@{.}[ur] \\ar@{.}[dr] & & \\\\\n      & & & X(2,1,2) & \\\\\n      & X(2) \\ar@{.}[uur] \\ar@{.}[ddr] & & & \\\\\n      & & & {*} & \\\\\n      & & {*} \\ar@{.}[ur] \\ar@{.}[dr] & & \\ldots \\\\\n      & & & {*} & \\\\\n                            }\n          }\n\\end{equation}\nThe notation is straightforward; for instance, by $X(1,2)$ is denoted\nthe DG module obtained by first performing the construction of Case (1),\nthen the construction of Case (2).  The rule for omitting nodes of the\ntree is that no $X(\\cdots)$  must contain two neighbouring digits $2$.\n\n\\begin{theorem}\n\\label{thm:Schmidt1}\nSuppose that $\\dim_k \\operatorname{H}\\!R \\geq 3$.  Then the AR quiver\n$\\Gamma$ of $\\sD^{\\operatorname{c}}(R)$ has infinitely many components.\n\\end{theorem}\n\n\\begin{proof}\nIt is a standing assumption in this section that $R$ is Gorenstein, so\neach component $C$ of $\\Gamma$ is isomorphic to ${\\mathbb Z} A_{\\infty}$ as a\nstable translation quiver by Theorem\n\\ref{thm:quiver_local_structure}(1).\n\nSince $\\dim_k \\operatorname{H}\\!R \\geq 3$, there exists an $e \\not\\in \\{ 0,d\n\\}$ such that $R$ has non-zero cohomology in degree $e$, so the above\nconstructions make sense.  Start with $X = R$ and consider the tree\n\\eqref{equ:tree}.  It follows from Lemma \\ref{lem:C}, (1) and (2),\nthat the function $\\varphi$ is constant with value $r$ on the $r$'th\ncolumn of the tree.  On the other hand, by Theorem\n\\ref{thm:quiver_local_structure}(3), the value of $\\varphi$ on the\n$n$'th horizontal row of a component $C \\cong {\\mathbb Z} A_{\\infty}$ of\n$\\Gamma$ is $n\\varphi_1$.  Hence, if the vertices corresponding to two\nmodules in the $r$'th column of the tree \\eqref{equ:tree} both belong\nto $C$, then they sit in the same horizontal row of vertices in $C$.\n\nEquation \\eqref{equ:j} implies that $\\operatorname{amp} \\tau Y = \\operatorname{amp} Y$ for each $Y$\nin $\\sD^{\\operatorname{c}}(R)$.  However, on $C$, the action of $\\tau$ is to move a\nvertex one step to the left.  It follows that the amplitude is\nconstant on each horizontal row of $C$.\n\nCombining these arguments, if the vertices corresponding to two\nmodules in the $r$'th column of the tree \\eqref{equ:tree}\nboth belong to $C$, then the modules have the same amplitude.\n\nOn the other hand, in the construction above, Case (1) makes the\namplitude grow by $d-1$ and Case (2) makes the amplitude grow by\n$e-1$.  Let $a_1, \\ldots, a_r$ be a sequence of the digits $1$ and $2$\nwhich does not contain two neighbouring digits $2$.  Suppose that the\nsequence contains $s$ digits $1$ and $r-s$ digits $2$.  Then since\n$\\operatorname{amp} X = \\operatorname{amp} R = d$ it holds that $\\operatorname{amp} X(a_1, \\ldots, a_r) = d +\ns(d-1) + (r-s)(e-1)$, and since $e < d$ it is clear that this value\nchanges when $s$ changes.  So by choosing $r$ sufficiently large, a\ncolumn of the tree \\eqref{equ:tree} can be achieved with an\narbitrarily large number of DG modules with pairwise different\namplitudes.\n\nBy the first part of the proof, this results in an arbitrarily large\nnumber of different components of $\\Gamma$, so $\\Gamma$ has infinitely\nmany components.\n\\end{proof}\n\n\\begin{theorem}\n\\label{thm:Schmidt2}\nSuppose that there is an $e$ with $\\dim_k \\operatorname{H}^e\\!R \\geq 2$.\nThen the AR quiver $\\Gamma$ of $\\sD^{\\operatorname{c}}(R)$ has families of distinct\ncomponents which are indexed by projective manifolds over $k$, and\nthese manifolds can be of arbitrarily high dimension.\n\\end{theorem}\n\n\\begin{proof}\nAgain, it is a standing assumption in this section that $R$ is\nGorenstein, so each component $C$ of $\\Gamma$ is isomorphic to ${\\mathbb Z}\nA_{\\infty}$ as a stable translation quiver by Theorem\n\\ref{thm:quiver_local_structure}(1).\n\nSet $X = R$.  With an obvious notation, consider\n$X(2_{\\alpha},1,2_{\\beta})$.  Then an isomorphism\n$X(2_{\\alpha},1,2_{\\beta}) \\cong\nX(2_{\\alpha^{\\prime}},1,2_{\\beta^{\\prime}})$ implies $X(2_{\\alpha},1)\n\\cong X(2_{\\alpha^{\\prime}},1)$ by Lemma \\ref{lem:C}(2), and then\n$\\beta = \\lambda\\beta^{\\prime}$ for a $\\lambda$ in $k$ by Lemma\n\\ref{lem:C}(3).  And $X(2_{\\alpha},1) \\cong X(2_{\\alpha^{\\prime}},1)$\nimplies $X(2_{\\alpha}) \\cong X(2_{\\alpha^{\\prime}})$ by Lemma\n\\ref{lem:C}(1), and then $\\alpha = \\kappa\\alpha^{\\prime}$ for a\n$\\kappa$ in $k$ by Lemma \\ref{lem:C}(3).\n\nThe $X(2_{\\alpha},1,2_{\\beta})$ hence give a family of pairwise\nnon-isomorphic objects of $\\sD^{\\operatorname{c}}(R)$ parametrized by the Cartesian\nproduct $\\{\\mbox{rays of $\\alpha$'s}\\} \\times \\{\\mbox{rays of\n  $\\beta$'s}\\}$.\n\nNow, $\\sup X = d$ so the class $\\alpha$ is in $\\operatorname{H}^{d-d+e}(X)$,\ncf.\\ the construction in Case (2).  However,\n\\[\n  \\operatorname{H}^{d-d+e}(X) = \\operatorname{H}^e(X) = \\operatorname{H}^e(R).\n\\]\nHence $\\{\\mbox{rays of $\\alpha$'s}\\} = {\\mathbb P}(\\operatorname{H}^e\\!R)$ where\n${\\mathbb P}$ denotes the projective space of rays in a vector space.\nMoreover, $\\sup X(2_{\\alpha},1) = d+(e-1)+(d-1) = 2d+e-2$ by Lemma\n\\ref{lem:C}, (1) and (2), so the class $\\beta$ is in\n$\\operatorname{H}^{(2d+e-2)-d+e}(X(2_{\\alpha},1))$.  However,\n\\begin{align*}\n  \\operatorname{H}^{(2d+e-2)-d+e}(X(2_{\\alpha},1))\n  & = \\operatorname{H}^{d+2e-2}(X(2_{\\alpha},1)) \\\\\n  & = \\operatorname{H}^{(d+e-1)+e-1}(X(2_{\\alpha},1)) \\\\\n  & \\cong \\operatorname{H}^e(R),\n\\end{align*}\nwhere $\\cong$ is by Lemma \\ref{lem:C}(1) because $\\sup X(2_{\\alpha}) =\nd+e-1$.  Hence it is also the case that $\\{\\mbox{rays of $\\beta$'s}\\}\n= {\\mathbb P}(\\operatorname{H}^e\\!R)$.\n\nThis shows that the $X(2_{\\alpha},1,2_{\\beta})$ give a family of\npairwise non-i\\-so\\-mor\\-phic objects of $\\sD^{\\operatorname{c}}(R)$ indexed by\n${\\mathbb P}(\\operatorname{H}^e\\!R) \\times {\\mathbb P}(\\operatorname{H}^e\\!R)$.  Note that the\nprojective space ${\\mathbb P}(\\operatorname{H}^{e}\\!R)$ is non-trivial since $\\dim_k\n\\operatorname{H}^e\\!R \\geq 2$.\n\nBy Lemma \\ref{lem:C}, (1) and (2), all the $X(2_{\\alpha},1,2_{\\beta})$\nhave the same value of $\\varphi$ (it is $4$), so if the vertices of\ntwo non-isomorphic ones belonged to the same component $C$ of\n$\\Gamma$, then they would be different vertices in the same horizontal\nrow of $C \\cong {\\mathbb Z} A_{\\infty}$ because the value of $\\varphi$ on the\n$n$'th row of $C$ is $n\\varphi_1$ by Theorem\n\\ref{thm:quiver_local_structure}(3).  However, it follows from\nEquation \\eqref{equ:j} that $\\inf(\\tau Y) = \\inf(Y) - d + 1$, so\ndifferent vertices in the $n$'th row of $C$ correpond to DG modules\nwith different $\\inf$, but the $X(2_{\\alpha},1,2_{\\beta})$ all have\nthe same $\\inf$ by Lemma \\ref{lem:C}, (1) and (2) (it is $0$).  Hence\nthe vertices of two non-isomorphic $X(2_{\\alpha},1,2_{\\beta})$'s must\nbelong to different components of $\\Gamma$, so a family has been found\nof distinct components of $\\Gamma$ parametrized by the projective\nmanifold ${\\mathbb P}(\\operatorname{H}^e\\!R) \\times {\\mathbb P}(\\operatorname{H}^e\\!R)$ over $k$.\n\nAn analogous argument with objects of the form\n$X(2_{\\alpha},1,2_{\\beta},1,\\ldots,1,2_{\\gamma})$ produces families of\ndistinct components of the AR quiver indexed by projective\nma\\-ni\\-folds of arbitrarily high dimension, as claimed.\n\\end{proof}\n\n\n\n\n\\section{Poincar\\'{e} duality spaces}\n\\label{sec:topology}\n\n\nThis section makes explicit the highlights of the previous sections\nfor DG algebras of the form $\\operatorname{C}^*(X;k)$.  The results first\nappeared in \\cite{artop}, \\cite{arquiv}, and \\cite{Schmidt}.\n\n\\begin{setup}\nIn this section, the field $k$ will have characteristic $0$.  By $X$\nwill be denoted a simply connected topological space with $\\dim_k\n\\operatorname{H}^*(X;k) < \\infty$.  Write\n\\[\n  n = \\sup \\{\\, i \\,|\\, \\operatorname{H}^i(X;k) \\neq 0 \\,\\}.\n\\]\n\nWhen the singular cochain complex $\\operatorname{C}^*(X;k)$ and singular\ncohomology $\\operatorname{H}^*(X;k)$ appear below, it is always with\ncoefficients in $k$, so I will use the shorthands $\\operatorname{C}^*(X)$ and\n$\\operatorname{H}^*(X)$.\n\\end{setup}\n\n\\begin{remark}\n\\label{rmk:A}\nThe singular chain complex $\\operatorname{C}^*(X)$ is a DG algebra under cup\nproduct, and by \\cite[exa.\\ 6, p.\\ 146]{FHTbook}, it is\nquasi-isomorphic to a commutative DG algebra $A$ satisfying the\nconditions of Setup \\ref{set:blanket}.\n\\end{remark}\n\n\\begin{remark}\nFor $X$ to be simply connected means that it is path connected and\nthat each closed path in $X$ can be shrinked continuously to a point.\nEquivalently, $X$ is path connected and its fundamental group\n$\\pi_1(X)$ is trivial.\n\nThe space $X$ is said to have Poincar\\'{e} duality over $k$ if there\nis an isomorphism\n\\[\n  \\operatorname{D}\\!\\operatorname{H}^*(X) \\cong \\Sigma^n \\operatorname{H}^*(X)\n\\]\nof graded left-$\\operatorname{H}^*(X)$-modules.  It is a classical theorem\nthat any compact $n$-dimensional manifold has Poincar\\'{e} duality;\nindeed, this is one of the oldest results of algebraic topology.\n\nA consequence of Poincar\\'{e} duality over $k$ is that there are\nisomorphisms of vector spaces\n\\[\n  \\operatorname{D}\\!\\operatorname{H}^i(X) \\cong \\operatorname{H}^{n-i}(X)\n\\]\nfor each $i$, and hence that the singular cohomology $\\operatorname{H}^*(X)$\nwith coefficients in $k$ is concentrated between dimensions $0$ and\n$n$ and has the same vector space dimension in degrees $i$ and $n-i$.\nGeometrically, this is in a sense the statement that the number of\nholes with $i$-dimensional boundary enclosed by $X$ is equal to the\nnumber of holes with $(n-i)$-dimensional boundary enclosed by $X$.\n\nAlgebraically, spaces with Poincar\\'{e} duality emulate Gorenstein\nalgebras; see \\cite{FHTpaper}.\n\\end{remark}\n\nFor the definition of $n$-Calabi-Yau categories, see Definitions\n\\ref{def:Serre} and \\ref{def:CY}.\n\n\\begin{theorem}\n\\label{thm:Chain_CY}\nThe following conditions are e\\-qui\\-va\\-lent.\n\\begin{enumerate}\n  \\item  $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X))$ is an $n$-Calabi-Yau category.\n  \\item  $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X)^{\\operatorname{o}})$ is an $n$-Calabi-Yau category.\n  \\item  $X$ has Poincar\\'{e} duality over $k$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nThis will involve showing that the conditions of the theorem are\nalso e\\-qui\\-va\\-lent to the following two conditions.\n\\begin{enumerate}\n  \\setcounter{enumi}{3}\n  \\item  $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X))$ has AR triangles.\n  \\item  $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X)^{\\operatorname{o}})$ has AR triangles.\n\\end{enumerate}\n\nFor the proof, $\\operatorname{C}^*(X)$ can be replaced with the commutative DG\nalgebra $A$ by Remark \\ref{rmk:A}.  So it is clear that\n(1)$\\Leftrightarrow$(2) and that (4)$\\Leftrightarrow$(5).\n\nCondition (3), that $X$ has Poincar\\'{e} duality, means\n${}_{\\operatorname{H}\\!A}(\\operatorname{D}\\!\\operatorname{H}\\!A) \\cong {}_{\\operatorname{H}\\!A}(\\Sigma^n \n\\operatorname{H}\\!A)$; since $A$ is commutative, Theorem\n\\ref{thm:Gorenstein}(2) implies that this is equivalent to $A$ being\nGorenstein.  Condition (4) is also equivalent to $A$ being Gorenstein\nby Theorem \\ref{thm:R}.  It follows that (3)$\\Leftrightarrow$(4).\n\n(1)$\\Rightarrow$(4) holds since a Calabi-Yau category has a Serre\nfunctor and hence AR triangles, see Definition \\ref{def:Serre},\nTheorem \\ref{thm:Serre}, and Definition \\ref{def:CY}.\n\n(3)$\\Rightarrow$(1).  The DG algebra $A$ is commutative, so Theorem\n\\ref{thm:Gorenstein}(3) implies that condition (3) is equivalent to\n\\[\n  \\operatorname{D}\\!A \\cong \\Sigma^n A\n\\]\nin the derived category of DG bi-$A$-modules.  Inserting this into\nEquation \\eqref{equ:Serre2} shows that the Serre functor of $\\sD^{\\operatorname{c}}(A)$\nis $\\Sigma^n$ so (1) holds, cf.\\ Definition \\ref{def:CY}.\n\\end{proof}\n\n\\begin{theorem}\n\\label{thm:Chain_Gamma}\nSuppose that $X$ has Poincar\\'{e} duality over $k$ and that it\nsatisfies $\\dim_k \\operatorname{H}^*(X) \\geq 2$.  Then each component of the\nAR quiver $\\Gamma$ of $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X))$ is isomorphic to ${\\mathbb Z}\nA_{\\infty}$.\n\nIf $\\dim_k \\operatorname{H}^*(X) = 2$, then $\\Gamma$ has $n-1$ components. \n\nIf $\\dim_k \\operatorname{H}^*(X) \\geq 3$, then $\\Gamma$ has infinitely many\ncomponents.\n\nIf $\\dim_k \\operatorname{H}^e(X) \\geq 2$ for some $e$, then $\\Gamma$ has\nfamilies of distinct components which are indexed by projective\nmanifolds over $k$, and these manifolds can be of arbitrarily high\ndimension.\n\\end{theorem}\n\n\\begin{proof}\nSince $\\operatorname{C}^*(X)$ is quasi-isomorphic to $A$, the theory of the\nprevious sections applies to $\\operatorname{C}^*(X)$.  As in the proof of\nTheorem \\ref{thm:Chain_CY}, since $X$ has Poincar\\'{e} duality,\n$\\operatorname{C}^*(X)$ is Gorenstein.  The present theorem hence follows from\nTheorems \\ref{thm:quiver_local_structure}, \\ref{thm:spheres},\n\\ref{thm:Schmidt1}, and \\ref{thm:Schmidt2}.\n\\end{proof}\n\nTheorem \\ref{thm:Chain_CY} and its proof imply that if $X$ has\nPoincar\\'{e} duality over $k$, then the AR quiver of\n$\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X))$ is a stable translation quiver.\n\n\\begin{theorem}\nThe AR quiver of $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X))$ is a weak homotopy invariant of\n$X$.\n\nIf $X$ is restricted to spaces with Poincar\\'{e} duality over $k$,\nthen the AR quiver of $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X))$, viewed as a stable\ntranslation quiver, is a weak homotopy invariant of $X$.\n\\end{theorem}\n\n\\begin{proof}\nIf $X$ and $X^{\\prime}$ have the same weak homotopy type, then by\n\\cite[thm.\\ 4.15]{FHTbook} there exists a series of quasi-isomorphisms\nof DG algebras linking $\\operatorname{C}^*(X)$ and $\\operatorname{C}^*(X^{\\prime})$.\nHence $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X))$ and $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X^{\\prime}))$ are\nequivalent triangulated categories, and this implies both parts of the\ntheorem.\n\\end{proof}\n\n\n\n\n\\section{Open problems}\n\\label{sec:open}\n\n\nLet me close the paper by proposing the following open problems.  The\nfirst one is due to Karsten Schmidt, see \\cite[sec.\\ 6]{Schmidt}.\n\n\\begin{problem}\nDevelop a theory of representation type of simply connected co\\-chain DG\nalgebras.\n\nWhat is known so far is the following.\n\\begin{enumerate}\n  \\item  By Theorem \\ref{thm:spheres}, if $\\dim_k \\operatorname{H}\\!R = 2$,\n    then the AR quiver $\\Gamma$ of $\\sD^{\\operatorname{c}}(R)$ has a finite number of\n    components. \n\n    Suppose that $R$ is Gorenstein.\n\n  \\item  By Theorem \\ref{thm:Schmidt1}, if $\\dim_k \\operatorname{H}\\!R \\geq\n    3$, then $\\Gamma$ has infinitely many components.\n    \n  \\item  By Theorem \\ref{thm:Schmidt2}, if $\\dim_k \\operatorname{H}^e\\!R \\geq\n    2$ for some $e$, then $\\Gamma$ has families of distinct components\n    which are indexed by projective manifolds, and these manifolds can\n    be of arbitrarily high dimension.\n\\end{enumerate}\nIt is tempting to interpret the DG algebras of (1) as having finite\nrepresentation type, and the ones of (3) as having wild representation\ntype.\n\nIf $\\dim_k \\operatorname{H}\\!R \\geq 3$ but $\\dim_k \\operatorname{H}^i\\!R \\leq 1$ for\neach $i$, then it is not clear whether the infinitely many components\nof $\\Gamma$ form discrete or continuous families, or indeed, what\nthese words precisely mean in the context.\n\nNote that some previous work does exist on the representation type of\nderived categories, see \\cite{GeissKrause}, but it does not apply to\nthe categories considered in this paper.\n\\end{problem}\n\n\\begin{problem}\nWhat is the structure of the AR quiver of $\\sD^{\\operatorname{c}}(R)$ if $R$ is not\nGorenstein?\n\nDo components of a different shape than ${\\mathbb Z} A_{\\infty}$ become\npossible?\n\\end{problem}\n\n\\begin{problem}\nGeneralize the theory to cochain DG algebras which are not simply\nconnected.\n\nPresently, not even the structure of $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(S^1;{\\mathbb Q}))$ is\nknown because $S^1$ and hence $\\operatorname{C}^*(S^1;{\\mathbb Q})$ is not simply\nconnected.\n\nA generalization to the non-simply connected case may impact on\nnon-com\\-mu\\-ta\\-ti\\-ve geometry for which more general cochain DG\nalgebras are being considered as vehicles.\n\\end{problem}\n\n\\begin{problem}\nIs there a link between the categories $\\sD^{\\operatorname{c}}(R)$ which have AR\nqui\\-vers consisting of ${\\mathbb Z} A_{\\infty}$-components, and the appearence\nof ${\\mathbb Z} A_{\\infty}$-components in representation theory?\n\nSee for instance \\cite[thm.\\ 4.17.4]{BensonI}.\n\\end{problem}\n\n\\begin{problem}\nIf a simply connected topological space $X$ has\n$\\dim_{{\\mathbb Q}}\\operatorname{H}^*(X;{\\mathbb Q}) = 2$, then it has the same rational\nhomotopy type as a sphere of dimension $\\geq 2$.  Theorem\n\\ref{thm:Chain_Gamma} implies that these are the only simply connected\nspaces with Poincar\\'{e} duality for which the AR quiver of\n$\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X;{\\mathbb Q}))$ has only finitely many components.\n\nIs this linked to any topological property which is special to these\nspaces?\n\\end{problem}\n\n\\begin{problem}\nLet $X$ and $T$ be topological spaces.  Suppose that $X$ is simply\nconnected with $\\dim_k \\operatorname{H}^*(X;k) < \\infty$, that $T$ has\n$\\dim_k \\operatorname{H}^i(T;k) < \\infty$ for each $i$, and let\n\\[\n  F \\rightarrow T \\rightarrow X\n\\]\nbe a fibration.  The induced morphism $\\operatorname{C}^*(X;k) \\rightarrow\n\\operatorname{C}^*(T;k)$ turns $\\operatorname{C}^*(T;k)$ into a DG\nleft-$\\operatorname{C}^*(X;k)$-module.  By \\cite[thm.\\ 7.5]{FHTbook} there is a\nquasi-isomorphism $k \\stackrel{\\operatorname{L}}{\\otimes}_{\\operatorname{C}^*(X;k)} \\operatorname{C}^*(T;k) \\simeq\n\\operatorname{C}^*(F;k)$, and this implies that if $\\dim_k \\operatorname{H}^*(F;k) <\n\\infty$ then $\\operatorname{C}^*(T;k)$ is an object of $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X;k))$.\n\nHence $\\operatorname{C}^*(T;k)$ corresponds to a collection of vertices with\nmultiplicities of the AR quiver $\\Gamma$ of $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X;k))$.\nIf $X$ has Poincar\\'{e} duality over $k$, then the theory of this\npaper gives information about the structure of $\\Gamma$, both locally\nand globally.\n\nDoes this have applications to the topological theory of fibrations?\n\nDo the structural results on $\\Gamma$ correspond to structural\nresults on topological fibrations?\n\\end{problem}\n\n\\begin{problem}\nBy considering the fibration $F \\rightarrow T \\rightarrow X$, looking\nat $\\operatorname{C}^*(T;k)$ as a DG left-$\\operatorname{C}^*(X;k)$-module, and using the\ntheory of this paper, one is in effect doing ``AR theory\nwith topological spaces''.\n\nIs there a way to do so directly with the spaces themselves?\n\\end{problem}\n\n\\begin{problem}\n\\label{prob:CY}\nIf $X$ is a topological space with $\\dim_k \\operatorname{H}^*(X;k) < \\infty$\nand Poincar\\'{e} duality over the field $k$ of characteristic $0$,\nthen $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X;k))$ is an $n$-Calabi-Yau category for some\n$n$ by Theorem \\ref{thm:Chain_CY}.  More generally, if $R$ is the DG\nalgebra from setup \\ref{set:blanket} and $R$ is commutative and\nGorenstein, then $\\sD^{\\operatorname{c}}(R)$ is a $d$-Calabi-Yau category.\n\nThese categories appear to behave quite differently from higher cluster\ncategories which are standard examples of Calabi-Yau categories.  For\ninstance, an $m$-cluster category contains an $m$-cluster tilting\nobject in terms of which every other object can be built in a single\nstep; this seems to be far from true for $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X;k))$ and\n$\\sD^{\\operatorname{c}}(R)$.\n\nWhich role do $\\sD^{\\operatorname{c}}(\\operatorname{C}^*(X;k))$ and $\\sD^{\\operatorname{c}}(R)$ play in the\ntaxonomy of Calabi-Yau categories?\n\nIn the context of Calabi-Yau categories, there is a ``Morita'' theorem\nfor higher cluster categories, see \\cite[thm.\\ 4.2]{KellerReiten}.  Is\nthere also a Morita theorem for the categories $\\sD^{\\operatorname{c}}(R)$?\n\\end{problem}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s:intro}\n\n\n\nIt has recently been shown that there is a common scaling relation\nbetween X-ray luminosity, radio luminosity and black hole mass in\nblack-hole X-ray binaries (BHXBs) and active galactic nuclei (AGN),\nstrongly suggesting that the accretion mechanism in both classes of\nobjects is identical \\citep{MHD03,FKM04}.  Given this universality of\naccretion physics, it is reasonable to assume that the different\nobservational states of BHXB should also exist in AGN\n\\citep*{Mei01,MGF03}. In this paper, I suggest a simple test looking\nfor observational consequences of such an equivalence of accretion\nstates.\n\nThe observational states of BHXBs are distinguished by different\nspectral shapes and, to some extent, different luminosities.  They are\nidentified with different physical modes of accretion onto a compact\nobject \\citep[see the review by][and references therein; these authors\n  suggest a non-standard state nomenclature which will be used\n  here]{MR04}.  One marked transition between accretion states occurs\nat Eddington ratios of order 1-10\\%, between the ``\\emph{hard X-ray,\n  steady jet}'' (conventionally called ``\\emph{low (luminosity)\/hard\n  (spectrum)}'') state, and the ``\\emph{thermal-dominant}''\n(``\\emph{high\/soft}'') state.  The difference between the \\emph{hard}\nand \\emph{thermal-dominant} observational states is ascribed to a\ndifference in the state of the underlying accretion disk around a\nstellar-mass black hole.  The \\emph{thermal dominant} state is\nidentified with radiatively efficient accretion through a standard\n\\citet{SS73} disk, while the \\emph{hard} state is identified with a\nradiatively inefficient accretion flow that replaces the innermost\npart of the standard disk at low accretion rates \\citep*[RIAF;\n  e.g.,][and references therein]{Esiea97}.  A second transition occurs\nat Eddington ratios of 20--30\\%, where the standard \\citet{SS73} thin\ndisk becomes unstable.  There, the spectrum changes from being\ndominated by thermal emission to showing a ``\\emph{steep power-law}''\n(\\emph{very high state}), i.e., an X-ray spectrum that is harder than\na pure blackbody but softer than that of the \\emph{hard} state.  This\ntransition may also be linked to a decrease in the radiative\nefficiency and hence a smaller thermal contribution to the total\nemission.  Alternatively, an increased contribution from inverse\nCompton scattering in a magnetically heated corona may be responsible\nfor the harder spectrum \\citep{HaMa91}.\n\nThe link between the state of the accretion disk and presence or\nabsence of a jet is an important part of the picture currently\nemerging for BHXBs and its analogy in AGN.  As implied by its name,\nthe \\emph{hard\/steady-jet} state is the only accretion state for which\ncontinuous, steady jets have been observed (and quite possibly are\nalways present).  When an object enters the \\emph{thermal-dominant}\nstate, the radio emission associated with the steady jet is quenched\n\\citep*[][e.g.]{GFP03}. Similarly, there are suggestions that the\nradio galaxies and quasars with powerful jets are analogues of\nblack-hole binaries in the \\emph{steep power-law} state \\citep*[or\nmaking the transition to that state; see][]{FBG04} which show\ntransient X-ray and radio flares, interpreted as discrete ejections of\nhigh-velocity jets \\citep{Mei01,GFP03}.\n\nWhile the detailed physics of accretion flows are subject of ongoing\nresearch and debate, the generic feature of the models for the\ntransition between the \\emph{hard} and \\emph{thermal dominant} state is a\nchange in the radiative efficiency at some critical value of $\\ensuremath{\\dot m}$,\nthe mass accretion rate \\ensuremath{\\dot M}\\ normalized to the Eddington accretion\nrate.  This transition occurs at low accretion rates $\\ensuremath{\\dot m} = \\ensuremath{\\dot m_{\\mathrm{crit}}} \\approx\n0.02$.  I show here that it is a consequence of such a state\ntransition that there is a change in the distribution of accretion\nflow luminosities at the critical accretion rate $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}$\n(\\S\\ref{s:theory}).  Using data from the literature, I examine whether\nthe distribution of observed AGN black hole masses and luminosities is\nconsistent with this aspect of a state transition in accretion flows\n(\\S\\ref{s:obs}).  The theoretical and observational picture of the\nsecond state transition, from \\emph{thermal dominant} to \\emph{steep power\nlaw}, is much less clear.  Therefore, I concentrate on the\nlow-efficiency transition in the first part of the paper. However, the\nexistence of this accretion state may be the key to resolving some\napparent contradictions of the simplest unification picture, which are\ndiscussed in \\S\\ref{s:disc}.  I conclude in \\S\\ref{s:conc}.\n\n\\section{Effect of a radiative efficiency change on the joint distribution of\nluminosities and black hole masses}\\label{s:theory}\n\nIn this section, I show that a transition from a radiatively\ninefficient accretion state (a RIAF in general) to one with much\nhigher radiative efficiency (a radiatively efficient accretion flow\n[REAF] such as a standard \\citealt{SS73} disk) is expected to change\nthe density of objects in the $(\\ensuremath{M_{\\mathrm{BH}}},\\ensuremath{L_{\\mathrm{bol}}})$ plane at the luminosity\ncorresponding to the transition accretion rate \\ensuremath{\\dot m_{\\mathrm{crit}}}.  I assume the\nsimplest possible generic model of an accretion state transition\noccurring at low radiative efficiency.  The notation used here follows\n\\citet{Esiea97}, but the argument is germane to any transition from\nRIAF to REAF, whatever the detailed physical theory of either state\nmay be.  Thus, even if the \\emph{form} of the radiative efficiency is\nin actuality different from that used here, the \\emph{method} of\nidentifying the difference between different accretion states by\nconsidering the distribution of objects in the $(\\ensuremath{M_{\\mathrm{BH}}},\\ensuremath{L_{\\mathrm{bol}}})$ plane is\nuniversally applicable.\n\nThe accretion flow radiates a fraction $\\zeta$ of the rest mass of\naccreted matter, so that the radiative bolometric luminosity \\ensuremath{L_{\\mathrm{bol}}}\\\nand the physical accretion rate \\ensuremath{\\dot M}\\ are related by\n\\begin{equation}\n\\ensuremath{L_{\\mathrm{bol}}} = \\zeta \\ensuremath{\\dot M} c^2\n\\label{eq:lbol_phys}\n\\end{equation}\nIn accretion disk models, the radiative efficiency does not depend\ndirectly on \\ensuremath{\\dot M}, but only on the dimensionless accretion rate $\\ensuremath{\\dot m}\n= \\dot M \/ \\ensuremath{\\dot M_{\\mathrm{Edd}}}$ \\citep[see, e.g.,][]{Cheea95,Esiea97}.  In this\ncase, using the accretion-rate dependent efficiency in\nEquation~\\ref{eq:lbol_phys} makes the definition of \\ensuremath{\\dot M_{\\mathrm{Edd}}}\\ circular,\nbecause \\ensuremath{\\dot M_{\\mathrm{Edd}}}\\ depends on the efficiency, which depends on \\ensuremath{\\dot m},\nwhich depends on \\ensuremath{\\dot M_{\\mathrm{Edd}}}.  Therefore, if the radiative efficiency varies\nwith accretion rate, it is necessary to define \\ensuremath{\\dot M_{\\mathrm{Edd}}}\\ for a\n\\emph{fixed} fiducial value of the radiative efficiency. This is\nusually chosen as $\\zeta_\\mathrm{fid}=0.1$ (\\citealt{NY95,Esiea97},\nbut compare \\citealt{Cheea95,BB99,MCF04}, e.g., who use\n$\\zeta_\\mathrm{fid}=1$; $\\ensuremath{\\dot m}=1$ corresponds to $\\ensuremath{L_{\\mathrm{bol}}}=\\ensuremath{L_{\\mathrm{Edd}}}$ if\n\\emph{and only if} the same value is used for the efficiency in the\ncalculation of $\\ensuremath{\\dot M}$ and $\\ensuremath{\\dot M_{\\mathrm{Edd}}}$ from $\\ensuremath{L_{\\mathrm{bol}}}$ and $\\ensuremath{L_{\\mathrm{Edd}}}$,\nrespectively).  \\ensuremath{\\dot M_{\\mathrm{Edd}}}\\ is then given by\n\\begin{eqnarray}\n\\ensuremath{\\dot M_{\\mathrm{Edd}}} & = & \\ensuremath{L_{\\mathrm{Edd}}}\/(0.1 c^2) \\nonumber\\\\\n      & = & 2.2\\times10^{-8} \\frac{\\ensuremath{M_{\\mathrm{BH}}}}{\\ensuremath{M_{\\sun}}}\\; \\ensuremath{M_{\\sun}}\n      \\mathrm{yr}^{-1}\n\\label{eq:theory.medd}\n\\end{eqnarray}\nFor all purposes other than calculating \\ensuremath{\\dot M_{\\mathrm{Edd}}}, I separate the\nconversion efficiency $\\zeta$ into $\\zeta = \\ensuremath{\\epsilon}\\,\\eta$.  Here, $\\eta$\nis the dissipation efficiency, the fraction of the rest mass energy of\naccreted matter that is liberated by the accretion process.  This\nenergy may either be radiated away, swallowed by the black hole as\nthermal or kinetic energy, or converted into kinetic energy of a disk\nwind or jets \\citep{NY95,BB99,FGJ03}.  The fraction that is radiated away is denoted by\n\\ensuremath{\\epsilon}. Since accreting matter has to lose the same binding energy in\nboth presumed accretion states, I assume that universally\n$\\eta=0.1$. The power liberated by the accretion flow is thus a\nconstant $P = 0.1 \\ensuremath{\\dot M} c^2$, while the radiated luminosity is given\nby\n\\begin{eqnarray}\n\\ensuremath{L_{\\mathrm{bol}}} & = & \\ensuremath{\\epsilon} \\ensuremath{\\dot m} \\, \\ensuremath{L_{\\mathrm{Edd}}} \\nonumber \\\\\n      & = & \\ensuremath{\\epsilon} \\ensuremath{\\dot m} \\frac{\\ensuremath{M_{\\mathrm{BH}}}}{\\ensuremath{M_{\\sun}}} \\, 1.26\\times 10^{31}\\,\\mathrm{W}. \\label{eq:defeps}\n\\end{eqnarray}\nIn the accretion state transition scenario, \\ensuremath{\\epsilon}\\ is a property of the\naccretion flow solution and therefore depends on \\ensuremath{\\dot m}: for $\\ensuremath{\\dot m}\n\\leq \\ensuremath{\\dot m_{\\mathrm{crit}}}$, \\ensuremath{\\epsilon}\\ is the low radiative efficiency of a RIAF, while\nat $\\ensuremath{\\dot m} > \\ensuremath{\\dot m_{\\mathrm{crit}}}$, it is the high efficiency of a REAF.\n\\citet{Esiea97} found $\\ensuremath{\\dot m_{\\mathrm{crit}}} = 0.08$ based on the ``strong ADAF''\nproposal \\citep{NY95}, while \\citet*{MLM00} predict $\\ensuremath{\\dot m_{\\mathrm{crit}}} = 0.02$\nbased on their coronal evaporation model.  Observations of black-hole\nbinaries find a value close to $\\ensuremath{\\dot m_{\\mathrm{crit}}} = 0.02$ \\citep{Mac03}, For pure\nnumerical convenience, we use $\\ensuremath{\\dot m_{\\mathrm{crit}}} = 0.01$ here; any conclusions\nthat would be affected by the difference between $\\ensuremath{\\dot m_{\\mathrm{crit}}} = 0.01$ and\n$\\ensuremath{\\dot m_{\\mathrm{crit}}} = 0.02$ would be on a weak footing anyway. Again ignoring\nphysical details, I assume the following simple form for the radiative\nefficiency as function of accretion rate:\n\\begin{equation}\n  \\ensuremath{\\epsilon} = \\left\\{ \\begin{array}{ll}\n     \\ensuremath{\\eps^{\\mathrm{lo}}} = 100 \\ensuremath{\\dot m} & \\mathrm{ if\\ } \\ensuremath{\\dot m} \\leq \\ensuremath{\\dot m}_{\\mathrm{crit}} \\\\\n     \\ensuremath{\\eps^{\\mathrm{hi}}} = 1  & \\mathrm{ if\\ } \\ensuremath{\\dot m}_{\\mathrm{crit}} < \\ensuremath{\\dot m}.\n\t \\end{array}\n    \\right.\n\\label{eq:valeps}\n\\end{equation}\nThe form of $\\ensuremath{\\eps^{\\mathrm{lo}}}$ follows \\citet{NY95}.  The numerical factor 100\nin the definition of \\ensuremath{\\eps^{\\mathrm{lo}}}\\ is chosen to make the luminosity\ncontinuous as \\ensuremath{\\dot m}\\ crosses \\ensuremath{\\dot m_{\\mathrm{crit}}}, since transitions between the\nspectral states can occur without a large change in luminosity.  As\nmentioned in the introduction, the accretion rate \\ensuremath{\\dot m}\\ should in\nprinciple be restricted to $\\ensuremath{\\dot m} < 0.3$ since the standard\n\\citet{SS73} thin disk becomes unstable and BHXBs are known to enter\nthe \\emph{steep power-law} state at higher accretion rates.  Lacking\naccurate knowledge of the radiative efficiency of that state, it\nappears more appropriate to concentrate on the detectability of the\nlow-accretion rate transition, which should not be influenced by the\npresence of this third accretion state at high accretion rates (see\n\\S\\ref{s:disc.superEdd}).\n\nThe assumed prescription for the radiative efficiency in\nEquation~\\ref{eq:valeps} ignores the finding that there is a\nhysteresis in the transition luminosity in at least five soft X-ray\ntransient binary systems, in the sense that the transition from the\n\\emph{hard} to the \\emph{thermal dominant} state occurs at a\nluminosity that is higher than a factor of about five than the reverse\ntransition (see \\citealp{MC03} and references therein;\n\\citealp{BO02}).  I will discuss the possible impact of this\nsimplification in \\S\\ref{s:disc.crit.parval}.\n\nIt is immediately obvious from equations \\ref{eq:defeps} and\n\\ref{eq:valeps} that for fixed black hole mass, the distribution of\nbolometric luminosities must show the same transition to a different\nscaling with \\ensuremath{\\dot m}\\ at $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}$ as \\ensuremath{\\epsilon}. Thus, this transition\nshould be detectable when considering the distribution of accreting\nsystems in the $(\\Mbh,\\Lbol)$ plane, as long as the distribution of accretion\nrates \\ensuremath{\\dot m}\\ does not similarly change precisely at \\ensuremath{\\dot m_{\\mathrm{crit}}}. This\nassumption is consistent with the common view that \\ensuremath{\\dot m}\\ is an\nobject-specific input parameter fixed by the surroundings of the\nparticular accreting system (e.g., the availability of gas in the AGN\nhost galaxy), while \\ensuremath{\\dot m_{\\mathrm{crit}}}\\ is a universal property of the accretion\nflow solution \\citep[although it is possible that the mass supply rate\n  at large radii does not in fact govern the accretion rate onto the\n  black hole; see][e.g.]{Pro05}.  Thus, I consider whether there any\nobservational evidence for such a change in the distribution of\nobserved AGN luminosities at fixed black hole mass.\n\n\\section{Observations}\\label{s:obs}\n\n\\subsection{Results}\n\n\\begin{figure}\n\\plotone{f1.eps}\n\\epsscale{1}\n\\caption{Distribution of AGN bolometric luminosities and black hole\n  masses (the symbol type indicates the method of black hole mass\n  determination as detailed below).  The diagonal solid lines are\n  lines of constant Eddington ratio $\\ensuremath{L_{\\mathrm{bol}}}\/\\ensuremath{L_{\\mathrm{Edd}}}$. They show the\n  bolometric luminosity as function of black hole mass from Equations\n  \\protect\\ref{eq:defeps} and \\protect\\ref{eq:valeps} for the given\n  values of the accretion rate \\ensuremath{\\dot m}. The radiative efficiency \\ensuremath{\\epsilon}\\\n  has a discontinuity at $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}=0.01$ which separates\n  radiatively inefficient flows (RIAF, $\\ensuremath{\\epsilon} = 100 \\ensuremath{\\dot m}$) at smaller\n  \\ensuremath{\\dot m}\\ from radiatively efficient flows (REAF, $\\ensuremath{\\epsilon} = 1$) at larger\n  \\ensuremath{\\dot m}.  The diagonal lines are equally spaced in \\ensuremath{\\dot m}, so that the\n  change in line spacing with respect to luminosity should be\n  reflected in a change in the density of objects in the $(\\Mbh,\\Lbol)$ plane\\ at\n  the line $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}$.  Contours and dots are measurements for the\n  12245 objects from the SDSS DR1 quasar catalog \\citep{SFHea03},\n  determined by \\citet{MD04} using the relation between optical\n  luminosity and size of the broad-line region (BLR) together with the\n  line width of the H\\,$\\beta$ or \\ion{Mg}{2} lines as virial mass\n  indicators.  These authors compute bolometric luminosities using a\n  fixed bolometric correction for the optical luminosity.  There are\n  12245 objects in total; the contours indicate the density of objects\n  in bins of 0.15\\,dex in both axes, starting at 1 object per bin and\n  increasing by a factor of $\\sqrt{2}$ per contour. The dashed line\n  indicates the FWHM cut $\\geq 2000$\\,km\\,s$^{-1}$ applied by\n  \\citeauthor{MD04} which excludes objects above and to the left of\n  this line. The solid line in the upper left-hand corner shows the\n  correlated change in \\ensuremath{L_{\\mathrm{bol}}}\\ and \\ensuremath{M_{\\mathrm{BH}}}\\ that an error in an object's\n  optical luminosity of 1\\,dex would produce.  Other symbols are those\n  black hole mass measurements from \\citet{WU02a} for which the\n  authors have determined the bolometric luminosity. The method of\n  black hole mass determination is indicated by the symbol type: solid\n  squares for resolved stellar kinematics (2 objects); open triangles\n  for reverberation mapping (36); plus signs for BLR size from optical\n  luminosity as in \\citeauthor{MD04} (139); and crosses for direct\n  measurements of the stellar velocity dispersion $\\sigma$ and the\n  \\ensuremath{M_{\\mathrm{BH}}}-$\\sigma$ relation (57).\n\\label{f:obs}}\n\\end{figure}\nFigure~\\ref{f:obs} shows the distribution of bolometric luminosities\nand black hole masses for the sample of objects from the compilation in\n\\citet{WU02a}, who list black hole masses obtained by a variety of\nmethods, and from \\citet{MD04}, who determine virial black hole masses\nfor objects from the Sloan Digital Sky Survey (SDSS) quasar catalog\n\\citep{SFHea03} using the virial method and the correlation between\nAGN luminosity and size of the broad-line region (BLR).  The SDSS\nspectroscopic quasar survey is an optically flux-limited survey of\nquasar candidates selected either from 5-color photometry or as\noptical point-source counterparts of sources detected in the FIRST\nradio survey \\citep[details of the selection are described\nin][]{qso_ts}.  The objects whose mass and luminosity are taken from\n\\citet{WU02a} are taken from a variety of samples and sources.  Most\nof the sources with black hole masses from the BLR size-luminosity\nrelation are radio-selected quasars, while most other AGN are Seyferts\nin nearby galaxies, or optically selected quasars from the Bright\nQuasar Survey (PG objects).\n\nThe diagonal lines in Fig.\\,~\\ref{f:obs} indicate the relation between\nluminosity and black hole mass for the indicated Eddington-scaled accretion\nrates $\\ensuremath{\\dot m}$ (equally spaced in $\\log \\ensuremath{\\dot m}$) and the\nresulting radiative efficiency from Equation~\\ref{eq:valeps}.  As\nargued above, the distribution of objects in $\\ensuremath{\\dot m}$ is not expected\nto change at $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}$. Therefore, the change in radiative\nefficiency at $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}$ should lead to a change in the density of\nobjects in the $(\\Mbh,\\Lbol)$ plane\\ at the line $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}$.  In\nFig.\\,\\ref{f:obs}, the density of objects does indeed change abruptly\nat this line.  In fact, nearly all objects from the samples used here\nlie \\emph{above} this line in the radiatively efficient regime, while\nonly a few objects have lower $\\ensuremath{\\dot m}$ as inferred from their black\nhole mass, luminosity, and the radiative efficiency prescribed by\nEquation~\\ref{eq:valeps}.\n\n\\begin{figure}\n\\plotone{f2.eps}\n\\caption{Bolometric luminosity against black hole mass for sources\n  listed by \\citet{MCF04}, including only sources with a measurement\n  of the bolometric luminosity (excluding upper limits).  The lines\n  for different Eddington-scaled accretion rates are identical to\n  those in Figure~\\ref{f:obs}.  Triangles indicate radio-loud quasars\n  (empty triangles for 3CR objects, filled for objects from\n  \\citealp{MD01} and \\citealp{HFHea02}), squares are FR~II radio\n  galaxies from the 3C sample, and circles are low-luminosity (mostly\n  FR~I) radio galaxies from the 3CR (empty) and B2 (filled) samples.}\n\\label{f:3cobs}\n\\end{figure}\nFigure\\,\\ref{f:3cobs} shows the equivalent of Fig.\\,\\ref{f:obs} for\nthe sample of radio-loud objects from \\citet*{MCF04}.  That sample\nincludes only objects with detected host galaxies and HST imaging,\nallowing the authors to measure the black hole mass from the\ncorrelation between host galaxy luminosity and black hole mass, as\nwell as the nuclear optical luminosity to determine the nuclear\nbolometric luminosity using a bolometric correction factor.  I have\nomitted sources with only an upper limit to the nuclear\nluminosity. Some of the objects classified as radio-loud quasars are\nalso included in the sample shown in Figure~\\ref{f:obs}. The accretion\nrate histogram derived from this figure (essentially, by a projection\nof this figure on a line perpendicular to the lines\n$\\ensuremath{\\dot m}=$\\emph{const}), \\citet{MCF04} shows a clear bimodality in the\naccretion rate distribution of AGN.  This bimodality is taken as\nstrong evidence for the presence of a transition between two accretion\nstates.  The presence of a ``gap'' in the distribution, roughly\nlocated at $3\\times10^{-4} < \\ensuremath{\\dot m} < 10^{-2}$, requires a\ndiscontinuous change of some property of the accretion flow (note that\nin contrast to the treatment here, \\citealt{MCF04} define $\\ensuremath{\\dot m}\n\\equiv \\ensuremath{L_{\\mathrm{bol}}}\/\\ensuremath{L_{\\mathrm{Edd}}}$, i.e., they use a constant radiative efficiency to\navoid having to make assumptions about the radiative efficiency as\nfunction of accretion rate).  Even though the accretion rate histogram\nclearly shows a bimodality, there is no obvious structure to the\ndistribution of the same objects in the $(\\Mbh,\\Lbol)$ plane.  In fact, it appears\npreferable to consider the full 2-dimensional distribution rather than\nits 1-dimensional projection, the accretion rate histogram, because\nthe impact of selection effects is more obvious in the 2-dimensional\ndiagram.  The impact of selection effects is discussed in detail in\n\\S\\ref{s:obs.selection} below.\n\nThe bulk of the 12245 objects from \\citet{MD04} lies in the region\nbetween $\\ensuremath{\\dot m}=0.1$ and $\\ensuremath{\\dot m}=1$ (contours in\nFig.\\,\\ref{f:obs}). These are so far from the line $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}$ that\nthese objects genuinely cannot be accreting in the\nlow-\\ensuremath{\\dot m}\\ radiatively inefficient mode.  This suggestion seems to be\nsupported by the fact that the vast majority of SDSS quasars has a\nblue bump \\citep[the average SDSS quasar spectrum clearly shows the\n  big blue bump; see][]{Berea01,YCVBea04}.  However, the implied\n\\ensuremath{\\dot m}\\ of many of these objects is so high that the \\citet{SS73}\nstandard disk solution does not hold any more and they should be in\nthe third accretion state.  We return to this point in the discussion\nsection below (\\S\\ref{s:disc.spec.higheff}).  The lack of\ninefficiently accreting objects in the SDSS sample is predominantly\ndue to the combined effect of the luminosity and flux limit for\nobjects in the SDSS quasar catalog, from which this sample is drawn\n(see \\S\\ref{s:obs.selection} below).  The lack of such objects\ntherefore does not allow to make any statement about the presence or\nabsence of two accretion modes in AGN.\n\nThe objects from \\citet{WU02a} include many Seyfert galaxies with\nluminosities that are much lower than the limit for inclusion in the\nSDSS quasar catalog, in particular among the objects with \\ensuremath{M_{\\mathrm{BH}}}\\ from\nstellar velocity dispersions (crosses in Fig.\\,\\ref{f:obs}) and those\nwith reverberation-mapping masses (triangles).  These objects are\npreferentially found above the line $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}=0.01$.  The lack of\nobjects with $\\ensuremath{\\dot m} < \\ensuremath{\\dot m_{\\mathrm{crit}}}$ at $\\ensuremath{M_{\\mathrm{BH}}} > 10^{9}\\ensuremath{M_{\\sun}}$ again is mainly\ndue to selection effects. But even at $\\ensuremath{M_{\\mathrm{BH}}} < 10^{9}\\ensuremath{M_{\\sun}}$, there are\nfewer objects with $\\ensuremath{\\dot m} < \\ensuremath{\\dot m_{\\mathrm{crit}}}$ compared to $\\ensuremath{\\dot m} > \\ensuremath{\\dot m_{\\mathrm{crit}}}$ (12\nbelow \\ensuremath{\\dot m_{\\mathrm{crit}}}\\ and 185 above).  The asymmetry is perhaps most\nsignificant among the objects with masses from host galaxy stellar\nvelocity dispersions, which are more likely to include low-luminosity\nAGN: only 11 of these are below \\ensuremath{\\dot m_{\\mathrm{crit}}}, compared to 46 above.  The\nsignificance of this difference is hard to quantify because the\nobjects listed by \\citet{WU02a} by construction do not constitute a\ncomplete sample.  But the small number of objects with $\\ensuremath{\\dot m} <\n\\ensuremath{\\dot m_{\\mathrm{crit}}}$ at $\\ensuremath{M_{\\mathrm{BH}}} < 10^{9}\\ensuremath{M_{\\sun}}$ may be taken at least as\nweak suggestive evidence for the presence of a radiatively inefficient\naccretion mode.\n\nIn stark contrast to the small number of low-efficiency objects in\nFig.~\\ref{f:obs}, the \\citet{MCF04} sample shown in\nFig.\\,\\ref{f:3cobs} shows a substantial number of objects in the\nlow-efficiency regime.  The key difference is that \\citet{MCF04} used\nonly optical observations and a bolometric correction to determine\nbolometric luminosities for these low-power radio galaxies, while\n\\citet{WU02a} explicitly excluded radio galaxies in their\ndeterminations of bolometric luminosities because ``obscuration and\nbeaming are significant'' in these sources.  By contrast,\n\\citet{MCF04} argue \\citep[based on the analysis by][]{CMSea02} that\nnuclear obscuration is not important in these low-power radio\ngalaxies, so that the nuclear optical luminosity can be used to infer\nthe bolometric luminosity.  This finding is disputed by \\citet{CR04}\n--- while they agree with \\citet{CMSea02} that the optical core flux\nof these objects is dominated by jet emission, they conclude that the\njet emission comes from scales larger than the obscuring material.\nThus, I refrain from drawing any conclusion from the large number of\nlow-luminosity (and hence low-accretion rate) sources in\nFigure\\,\\ref{f:3cobs}, since they would hinge on which assumptions are\nadopted about the nature of the nuclei of low-luminosity radio\ngalaxies.\n\nI next consider the impact of the other assumptions going into\nconstruction of the distributions in the $(\\Mbh,\\Lbol)$ plane\\ on the results\nobtained here.\n\n\\subsection{Impact of assumptions}\n\\label{s:obs.assumptions}\n\nIn order to assess the significance of any conclusions about the\npresence or absence of two accretion modes, it is necessary to\nconsider the impact of the assumptions going into the determination of\nthe distribution of objects in the $(\\Mbh,\\Lbol)$ plane.  Perhaps the most serious\ncaveat has already been mentioned above: the accretion rate\n\\ensuremath{\\dot m}\\ might not necessarily be the same as the mass supply rate, but\ncould be set by processes internal to the accretion disk\n\\citep{Pro05}, so that the value of \\ensuremath{\\dot m}\\ may be a property of the\naccretion disk solution rather than an independent parameter set by\nthe availability of gas supply in the host galaxy, as has been assumed\nhere. If this is the case, the distribution of objects in the\n$(\\Mbh,\\Lbol)$ plane\\ would still provide information about the physics of\naccretion disks if the mass supply rate could be determined\nindependently.\n\nA detailed discussion in \\S\\ref{s:obs.err} of the appendix shows that\nerrors in the determination of black hole mass and bolometric\nluminosities would either blur the transition region or systematically\nshear the distribution of points in the $(\\Mbh,\\Lbol)$ plane, making the detection\nof the transition more difficult, but not impossible.  In\n\\S\\ref{s:obs.lum} of the appendix, I conclude that the bolometric\nluminosity constitutes the most reliable measure of accretion\nluminosity.  However, large systematic errors are introduced in\nbolometric luminosities by deriving bolometric corrections from one\nsample \\citep[the objects in the library of AGN SEDs by][]{EWMea94}\nbut applying them to the SDSS quasar sample, which goes beyond\ntraditional UV excess selection and includes objects with much redder\noptical colors, and therefore possibly different SEDs in other\nwavelength regions, than the samples on which the bolometric\ncorrections are based.\n\nIn the remainder of this section, I consider in detail the impact of\nassumptions about the values for critical accretion rate and radiative\nefficiency, and of the sample selection function on this distribution.\nI will then discuss the implications of the observed distributions.\n\n\\subsection{Values for critical accretion rate and radiative\nefficiencies}\\label{s:disc.crit.parval}\n\nIn order to determine the accretion rate \\ensuremath{\\dot m}\\ from the observable\n\\ensuremath{L_{\\mathrm{bol}}}, it was necessary to assume a value for the fraction of the\naccreted matter's rest-mass energy that is dissipated by the accretion\nprocess, and to assume radiative efficiencies for the different\naccretion states.  In addition, the location of the line dividing\nefficient from inefficient accretors in the $(\\Mbh,\\Lbol)$ plane\\ obviously\ndepends on the value of \\ensuremath{\\dot m_{\\mathrm{crit}}}.\n\nHere, the fraction of rest-mass energy that is liberated as accretion\npower was taken to be $\\eta=0.1$ universally.  However, this value in\nfact depends on the location of the last stable orbit around the black\nhole, which in turn depends the black hole spin.  For a non-rotating\nblack hole (Schwarzschild metric), $\\eta=0.06$.  The efficiency of\naccretion on a rotating black hole (Kerr metric) can be lower than the\nSchwarzschild value if it is counter-rotating with respect to the\ndisk, while a maximally co-rotating black hole would result in\n$\\eta=0.42$.  As there is likely a large variation of the black hole\nspin from AGN to AGN based on its particular accretion history, the\nvalue of $\\eta$ will also differ from object to object.  The value of\n$\\eta$ changes the location of the lines $\\ensuremath{\\dot m} = \\mathrm{const.}$ in\nFig.\\,\\ref{f:obs}, i.e., there would be a different set of these lines\nfor each value of $\\eta$.  The appropriate simplification would be to\nuse the ensemble average of $\\eta$ to draw this set of lines; since an\naccurate measurement of $\\eta$ is not available for any object, use of\nthe conventional value $\\eta=0.1$ appears most appropriate.\n\nI have chosen \\ensuremath{\\dot m_{\\mathrm{crit}}}, \\ensuremath{\\eps^{\\mathrm{lo}}}, and \\ensuremath{\\eps^{\\mathrm{hi}}}\\ to match observations of\nblack-hole binaries approximately.  If the true values of \\ensuremath{\\dot m_{\\mathrm{crit}}}, \\ensuremath{\\eps^{\\mathrm{lo}}},\nand \\ensuremath{\\eps^{\\mathrm{hi}}}\\ were smaller than assumed here, the transition between\nefficient and inefficient accretion would occur at smaller\nluminosities, where there are not many objects in Fig.\\,\\ref{f:obs}.\nIn this case, \\emph{all} objects in Fig.~\\ref{f:obs} might in fact be\nin the efficient state, and the luminosity range occupied by\ninefficiently accreting objects has not yet been reached.\n\nAnother complication arises from the fact that there is be a\nhysteresis loop between the two accretion states in black-hole\nbinaries, with the low-high transition occurring at a different\naccretion rate than the high-low transition \\citep[][and references\ntherein]{MC03}. These authors find that the transition back to the\nlow-efficiency hard state occurs at a luminosity $\\sim5$ times lower\nthan the hard-to-soft transition from low to high efficiency, so that\nobjects with identical luminosity can have different accretion rates.\nIf AGN can undergo an accretion state change during their active\nphase, this would apply to them, too, so that the instantaneous\nluminosity would not reveal the instantaneous accretion rate.  The\nadopted value of \\ensuremath{\\dot m_{\\mathrm{crit}}}\\ would again need to be interpreted as an\nensemble average.\n\nIn all these cases, the transition between efficient and inefficient\naccretion would still be visible in the $(\\Mbh,\\Lbol)$ plane, but the transition\nwould be blurred by the scatter of true efficiencies about the assumed\naverage value, or the location of the transition would be moved. In\nthis case, it might occur in a region that currently does not contain\nany objects.  These difficulties can therefore be overcome with\nobservations of more objects.\n\n\\subsection{Ignoring a possible third accretion state}\\label{s:disc.superEdd}\n\nEven if the parameters chosen for the low-accretion rate transition\nare correct, there is a third \\emph{super-Eddington} accretion mode\nwith a different efficiency of the standard thin disk. In fact, the\nstandard thin-disk model becomes self-inconsistent and must be\nmodified to include advection above $\\ensuremath{\\dot m}=$0.2--0.3 \\citep[e.g.]{KB99},\ni.e., they become radiatively inefficient, too.  Accretion at even\nhigher rates could become even more radiatively inefficient\n\\citep[see][e.g.]{BM82,Abr04}.  ``Super-Eddington'' would then refer\nto the fact that the accretion rate exceeds the Eddington \\emph{rate}\nfor the fiducial efficiency, while the radiative \\emph{luminosity} may\nstill be \\emph{below} the Eddington luminosity because of the decrease\nin the actual radiative efficiency.  Consideration of this accretion\nstate is particularly relevant since BHXBs enter a \\emph{steep\npower-law} state at roughly 30\\% of the Eddington luminosity, with a\nharder X-ray spectrum than the thermal-dominated state.\n\nThe physics of these high-accretion rate inefficient disks are the\nsubject of ongoing research.  No universally accepted predictions are\navailable yet for their radiative efficiency and possible\ntime-dependent behavior.  Generically, though, the radiative\nefficiency must be dropping with increasing \\ensuremath{\\dot m}\\ just above this\nsecond critical accretion rate. The accretion luminosity would\ntherefore increase more slowly than \\ensuremath{\\dot m}. It could even\n\\emph{decrease} with increasing \\ensuremath{\\dot m}\\ if the accretion efficiency\nscales like $\\ensuremath{\\dot m}^p$ with $p<-1$ in the transition region.  The same\nradiative luminosity may then be produced either by a standard\naccretion flow with modest \\ensuremath{\\dot m}\\ and high \\ensuremath{\\epsilon}, or by a\nsuper-Eddington flow with high \\ensuremath{\\dot m}\\ and modest \\ensuremath{\\epsilon}.  In any case, a\nsecond transition to a low-efficiency state would lead to a ``pileup''\nof sources in the $(\\Mbh,\\Lbol)$ plane\\ near the super-Eddington \\ensuremath{\\dot m_{\\mathrm{crit}}}. In this\ncontext, it may be of relevance that the contours for the SDSS quasars\nin Fig.~\\ref{f:obs} are peaked around the line $\\ensuremath{\\dot m}=0.3$.\n\nIn an extreme case, the radiative efficiency of the super-Eddington\nstate could be so low that the luminosity of objects with \\emph{any}\n\\ensuremath{M_{\\mathrm{BH}}}\\ can be produced in a radiatively inefficient flow. In this case,\nthere may in fact be \\emph{no} standard radiatively efficient\naccretion flows at all \\citep[cf.][]{Cheea95}.  Unless this is the\ncase, this high-Eddington rate transition at $\\ensuremath{\\dot m}=0.2$--0.3 should\nnot influence the detectability of the low-Eddington rate transition at\n$\\ensuremath{\\dot m_{\\mathrm{crit}}}\\approx 0.01$.\n\n\\subsection{Impact of sample selection effects}\n\\label{s:obs.selection}\n\nThe selection of a sample of AGN for which black hole masses and\nabsolute luminosities are to be determined always restricts the\nresulting distribution of objects to certain regions of the\n$(\\Mbh,\\Lbol)$ plane. This can happen both explicitly, by a luminosity or line\nwidth cut, e.g., or implicitly, because the distribution of AGN\nluminosities and black hole masses over redshift systematically\nremoves objects from flux-limited surveys, e.g.  The correct\ninterpretation of the observed distribution of objects in the\n$(\\Mbh,\\Lbol)$ plane\\ requires to take into account these selection effects.\n\n\n\\subsubsection{Selection effects in \\citet{MCF04} sample}\n\\label{s:obs.sel.mcf}\n\n\\begin{figure}\n\\plotone{f3.eps}\n\\epsscale{1}\n\\caption{\\label{f:MCFmdotz}Dimensionless accretion rate \\ensuremath{\\dot m}\\ against\n redshift $z$ for the \\citet{MCF04} objects shown in\n Figure\\,\\ref{f:3cobs}; for consistency with that work, this figure\n defines $\\ensuremath{\\dot m} \\equiv \\ensuremath{L_{\\mathrm{bol}}}\/\\ensuremath{L_{\\mathrm{Edd}}}$, i.e., with a constant radiative\n efficiency.  Symbols as in Figure\\,\\ref{f:3cobs}: triangles for radio-loud\n quasars, empty triangles for 3CR and filled from two other samples;\n squares for 3CR FR~II radio galaxies; circles for low-luminosity\n radio galaxies from 3CR (empty) and B2 (filled).  The double arrow\n indicates the location of the gap identified by \\citet{MCF04} in the\n accretion rate histogram.  The dashed lines show the maximum value of\n \\ensuremath{\\dot m}\\ and $z$ for the set of low-luminosity radio galaxies (open\n circles).  These constitute the majority of low-\\ensuremath{\\dot m}\\ objects, but\n are drawn from a completely different volume than the 3CR radio-loud\n quasars (open triangles), which constitute the majority of\n high-\\ensuremath{\\dot m}\\ objects: the former are limited to $z\\leq 0.29$ and\n $\\ensuremath{\\dot m} \\leq 1.95\\times10^{-3}$, while the latter are found at\n $z\\geq0.3$ and $\\ensuremath{\\dot m} > 10^{-2}$.  Figure\\,\\ref{f:MCFhisto} below\n compares the accretion rate histograms of the full sample, and of a\n revised sample which includes only objects at $z\\leq 0.29$.}\n\\end{figure}\n\\begin{figure}\n\\plotone{f4.eps}\n\\caption{\\label{f:MCFhisto}Accretion rate histogram for \\citet{MCF04}\n  objects, excluding objects with upper limits on the bolometric\n  luminosity.  The open histogram shows the histogram of \\ensuremath{\\dot m}\\ for\n  the entire remaining sample, while the shaded histogram shows the\n  histogram for objects at $z\\leq0.29$, which is the highest redshift\n  of any source with $\\ensuremath{\\dot m} \\la 2\\times10^{-3}$.  Thus, the shaded\n  histogram excludes those sources located in the volume in which\n  low-\\ensuremath{\\dot m}\\ sources are inaccessible to observations.  This\n  ``equal-redshift-range'' sample does not show a pronounced gap in the\n  accretion rate distribution any more. }\n\\end{figure}\nThe selection effects of the \\citet{MCF04} sample of objects are the\nsimplest, as they use mostly objects from two complete flux-limited\nradio surveys, the 3CR and B2 surveys.  The 3CR \\citep{LRL83,SMAea85}\nis a 178\\,MHz survey of 13,920 square degrees of sky to 10.9\\,Jy. It\nincludes a mix of low-power FRI (radio luminosity at 178\\,MHz in the\nrange $10^{24} < \\ensuremath{L_{\\mathrm{178}}} < 10^{28}$\\,W\\,Hz$^{-1}$) and higher-power FRII\nradio galaxies ($10^{25} < \\ensuremath{L_{\\mathrm{178}}} < 10^{28}$\\,W\\,Hz$^{-1}$) at $z\\leq\n0.3$, as well as radio-loud quasars at $z\\geq 0.3$, with luminosities\nin the range $10^{27} < \\ensuremath{L_{\\mathrm{178}}} < 10^{29}$\\,W\\,Hz$^{-1}$.  The B2 survey\n\\citep{CFFea75_III} is a survey of about one-half the area of the 3C\nwith a flux limit of 0.2--0.25\\,Jy and a magnitude limit $m_V<16.5$ on\nthe galaxies identified optically with the radio sources, i.e., it\nsamples a smaller volume than the 3C.  Therefore, it predominantly\nincludes low-luminosity objects (high-luminosity objects have a lower\nvolume density and therefore require a large-volume sample to be\nfound): the B2 objects used by \\citet{MCF04} are all at $z\\leq 0.15$\nand have $10^{22} < \\ensuremath{L_{\\mathrm{178}}} < 10^{26}$\\,W\\,Hz$^{-1}$.  The sample is\naugmented by further radio-loud quasars in the redshift range $0.15\n\\leq z \\leq 0.3$.  Thus, the total sample has substantial overlap in\n\\ensuremath{L_{\\mathrm{178}}}\\ and redshift.  However, Figure\\,\\ref{f:MCFmdotz} shows that the\nmajority of high-accretion rate objects are drawn from a completely\ndifferent volume than all the low-accretion rate objects.\nLow-accretion rate objects, in particular the low-luminosity radio\ngalaxies from the B2 sample, are exclusively found at $z\\leq 0.29$,\nwhile \\emph{all} of the 3C radio-loud quasars are at $z > 0.3$. The\nonly high-accretion rate objects at $z < 0.3$ included in the\n\\citet{MCF04} histogram are the 3C radio galaxies (i.e., objects with\nboth powerful radio jets as well as broad emission lines and some\nnon-stellar continuum, but with lower optical luminosities than\nradio-loud quasars) as well as the radio-loud quasars from\n\\citet{MD01}.\n\nFigure\\,\\ref{f:MCFhisto} shows the effect of removing all sources at\n$z>0.3$ from the sample, where objects with $\\ensuremath{\\dot m} \\la 10^{-2}$ are\ninaccessible to the surveys used.  The high-\\ensuremath{\\dot m}\\ peak in the\naccretion rate histogram disappears nearly completely, and moves from\n$\\ensuremath{\\dot m}_\\mathrm{peak} \\approx 0.1$ to $\\ensuremath{\\dot m}_\\mathrm{peak} \\approx\n0.01$.  There is no clear bimodality any more.  Thus, not only is the\ngap in the accretion rate distribution not obvious when considering\nthe full 2-dimensional distribution of objects from \\citet{MCF04} in\nthe $(\\Mbh,\\Lbol)$ plane\\ (Fig.\\,\\ref{f:3cobs}), but the gap is an artifact of\nselection effects: high-\\ensuremath{\\dot m}\\ objects are predominantly found at\n$z\\ga 0.3$ in the samples used, while low-\\ensuremath{\\dot m}\\ objects are\nrestricted to $z\\la 0.3$.  Furthermore, even after applying an\nidentical redshift cut to the samples used by \\citet{MCF04}, the\n\\emph{volumes} sampled to find the different subsets of objects are\nstill different because of the different areas of sky and that have\nbeen surveyed by the B2 and the 3C survey, and because of the optical\nflux limit imposed on B2 galaxy identifications.\n\n\\subsubsection{Selection effects in the SDSS quasar sample}\n\nThe same criticism (objects with different accretion rates being drawn\nfrom different volumes) also applies, of course, to the objects shown\nin Figure\\,\\ref{f:obs}.  The objects for which \\citet{MD04} determine\nblack hole masses are drawn from the SDSS DR1 quasar catalog\n\\citep{SFHea03}. This catalog is restricted in the $i$-band, both by\nflux limit and by a limit in absolute magnitude $M_i \\leq -22$, which\nroughly corresponds to $\\ensuremath{L_{\\mathrm{bol}}} \\geq 10^{38}$\\,W (using the relation\nbetween \\ensuremath{L_{\\mathrm{bol}}}\\ and $M_B$ given by \\citealt{MD04} and approximating\nquasar spectra as a power law with $f_\\nu \\propto \\nu^{-0.5}$,\nresulting in $M_B - M_i = 0.3$).  This luminosity cut means that\nobjects with $\\ensuremath{\\dot m}\\leq\\ensuremath{\\dot m_{\\mathrm{crit}}} = 0.01$ are only included in this quasar\ncatalog if they satisfy $\\ensuremath{M_{\\mathrm{BH}}} \\ga 10^9\\ensuremath{M_{\\sun}}$.  The flux limit of the\nquasar survey restricts the detection of such objects to $z \\la 0.33$\nfor $ugri$-selected objects ($m_i \\leq 19.1$) and $z \\la 0.53$ for\n$griz$-selected objects ($m_i \\leq 20.2$).  The surveyed volume is\ntherefore limited, and hence the number of such objects that can be\nfound by the SDSS.\n\nAt higher redshifts, the surveyed volume is much larger, but now the\nflux limit introduces a redshift-dependent luminosity limit.  As\ndiscussed by \\citet{MD04}, the use of the luminosity in the black hole\nestimation in a flux-limited implies a correlation between black hole\nmass and redshift, leading to an effective lower limit on the black\nhole mass as a function of redshift.  Thus, the SDSS quasar survey\nmisses high-mass black holes with low luminosities.  Furthermore,\n\\citet{MD04} include only objects with $v_\\mathrm{FWHM} \\geq\n2000$\\,km\\,s$^{-1}$ in their sample, which excludes objects to the\nleft and above the dashed line.  This cut is equivalent to a\nmass-dependent upper limit on \\ensuremath{\\dot m}\\ for objects in the sample.\n\nIn general, the volume density of faint AGN (more than 3 magnitudes\nfainter than those observed by the SDSS) increases towards redshifts\naround 2 in the same way as that of luminous quasars \\citep{WWBea03},\ni.e., faint AGN outnumber bright ones at all redshifts.  We are\ntherefore missing black hole mass determinations for a substantial\nfraction of the AGN population, with a strong bias against\nlow-luminosity AGN at high redshifts.\n\n\\subsubsection{Selection effects for remaining sources}\n\nThe remainder of the objects have unquantifiable selection effects,\nsince they are not drawn from samples with well-defined selection\ncriteria.  However, \\citet{WU02a} give a detailed discussion of\npossible selection effects in the construction of a diagram such as\nFig.\\,\\ref{f:obs}.  The important incompleteness for the present\ndiscussion is again the lack of objects with high black hole masses\n($\\ensuremath{M_{\\mathrm{BH}}} \\geq 10^8\\ensuremath{M_{\\sun}}$) and low luminosities ($\\ensuremath{L_{\\mathrm{bol}}} \\leq\n10^{38}$\\,W).  Since black hole mass correlates with the bulge\nluminosity of the host galaxy \\citep[][and references therein]{MH03},\na fraction of these objects is missing from AGN samples at all\nredshifts: a low-luminosity AGN is more difficult to detect against\nthe more luminous host galaxy, and the small volume of the\nlow-redshift universe means that high-mass black holes are rare in it.\nThere is an additional observer bias towards high-luminosity AGN:\nblack hole masses are typically first determined for objects drawn\nfrom the brightest samples, such as the 3C radio survey, the Bright\nQuasar Survey, or the SDSS quasar survey, even though faint AGN are\nmore numerous.\n\nFurthermore, there is some natural practical bias towards a certain\ntype of AGN for a certain method of black hole mass determination. For\nexample, the use of the stellar velocity dispersion method is only\npossible for AGN which are sufficiently nearby and of low luminosity\nto allow observation of the host galaxy.  A sample that exclusively\nuses this method will therefore be biased against high-luminosity AGN\nwith low-mass black holes, because these reside in low-luminosity\nhosts which are likely to be outshone by the nucleus.  Selection\neffects of this kind can be avoided by making use of the broadest\npossible range of black hole mass determination methods.\n\nIn conclusion, the sample selection function severely influences the\nobserved distribution of objects in all regions of the $(\\Mbh,\\Lbol)$ plane. In\nparticular, flux limits on AGN surveys (both explicit flux limits in\ncomplete surveys, and implicit flux limits because the faintest AGN\nare practically unobservable) bias any survey against the inclusion of\nlow-luminosity objects, whose black hole masses are crucial in\ndetecting the expected difference between efficiently and\ninefficiently accreting objects.  Finally, there is a population of\nobscured AGN that might be at least as numerous as known AGN\npopulations \\citep{RE04,TUCea04}, but not many black hole mass\nmeasurements exist for Type 2 AGN, and none exist for obscured sources\nat high redshift.\n\nThus, considering distributions like those in Figure\\,\\ref{f:obs} may\nyield a misleading picture because every data point obtains an equal\nweight, regardless of the volume that had to be surveyed to find it,\nand regardless of the properties of objects that are excluded by\nselection effects.  The effect of removing sources at high redshift,\nwhere low-luminosity AGN are not included in flux-limited samples,\nfrom Fig.\\,\\ref{f:obs} would be similar to the effect on the accretion\nrate histogram in that it removes the bulk of objects which are\nclearly at $\\ensuremath{\\dot m}>0.01$.  The more sound way to perform the test\nsuggested here will be to determine the volume density of objects with\na given \\ensuremath{L_{\\mathrm{bol}}}\\ and \\ensuremath{M_{\\mathrm{BH}}}, preferably in a volume-limited sample.\n\nHowever, the evidence cited above in favor of the existence of a\ntransition relied mostly on the asymmetry in the $\\ensuremath{\\dot m}$ distribution\nof objects with masses from host galaxy stellar velocity\ndispersions. Since these are more likely to include low-luminosity AGN\nthan flux-limited quasar surveys, this conclusion is not affected\nseverely by the selection effects discussed in this subsection.\n\n\n\\section{Discussion}\n\\label{s:disc}\n\nThe main aim of this paper is to point out that a transition in the\nradiative efficiency at some critical accretion rate \\ensuremath{\\dot m_{\\mathrm{crit}}}\\ should\nlead to a change in the distribution of AGN in the $(\\Mbh,\\Lbol)$ plane, and that\nthis prediction is testable with presently available techniques.  If\nthe analogy between AGN and black-hole binaries is complete, the\ndifference between the different accretion states should not only be\nevident in the $(\\Mbh,\\Lbol)$ plane, but also manifest itself in a spectral\ndifference between efficiently and inefficiently accreting\nobjects. This section considers the spectral evidence for an analogy\nof AGN and black-hole binary accretion states, first for the 12\nlow-\\ensuremath{\\dot m}\\ objects identifiable in Fig.\\,\\ref{f:obs} and then for the\nremaining objects which are presumably in a high-efficiency accretion\nstate. I also discuss whether the spectral and jet properties of AGN\nin general fit into the same classification scheme as black-hole\nbinaries.  Before considering these points, I briefly discuss the\nevidence for the presence of a transition in the accretion properties\nof radio-loud AGN presented by \\citet{MCF04}.\n\n\\subsection{A bimodality in the \\ensuremath{\\dot m}\\ distribution of AGN with\npowerful jets?}\n\\label{s:disc.3ctransition}\n\nIn \\S\\ref{s:obs.sel.mcf} above, I showed that the obvious bimodality\nin the accretion rate histogram of radio-loud AGN presented by\n\\citet{MCF04} is to a large degree a consequence of sample selection\neffects, with high-\\ensuremath{\\dot m}\\ objects (high-efficiency accretors) being\ndrawn from a much larger volume than those with low\n\\ensuremath{\\dot m}\\ (low-efficiency accretors).  This does not, however, imply that\nthe conclusions drawn by \\citet{MCF04} are necessarily invalid ---\n\\S\\ref{s:obs.sel.mcf} merely shows that selection effects are\nresponsible at least in part for the apparent bimodality, but not that\nthere is no such bimodality, even though Figure~\\,\\ref{f:3cobs} shows\nno obvious gap in the distribution of radio-loud objects in the\n$(\\Mbh,\\Lbol)$ plane.  To clarify this issue, a sample of AGN is needed which\nincludes \\emph{all} objects with \\ensuremath{\\dot m}\\ in the range under\nconsideration.\n\n\\subsection{Spectral evidence for inefficient accretion?}\\label{s:disc.spec.loweff}\n\nWe concluded above that the small number of objects in the sample of\nAGN shown in Figure~\\ref{f:obs} with $\\ensuremath{\\dot m} < \\ensuremath{\\dot m_{\\mathrm{crit}}}$ at $\\ensuremath{M_{\\mathrm{BH}}} <\n10^{9}\\ensuremath{M_{\\sun}}$ may be taken as weak suggestive evidence for the presence\nof a radiatively inefficient accretion mode.  The 12 objects which\nhave $\\ensuremath{M_{\\mathrm{BH}}} < 10^{9}\\ensuremath{M_{\\sun}}$ and $\\ensuremath{\\dot m} < \\ensuremath{\\dot m_{\\mathrm{crit}}}=0.01$ are Mrk~3 and 270,\nand NGC 513, 1052, 2110, 2841, 3786, 3998, 4258, 4339, 5929, and 6104;\nNGC~4258 has a direct black hole mass measurement from water maser\nkinematics, while the black hole masses for the remainder have been\ndetermined from the stellar velocity dispersion of the host galaxy and\nthe $M-\\sigma$ relation.  Is there any spectral evidence that these\nare in a low-efficiency accretion state?  If the black-hole binary\nscheme applied in complete analogy to AGN, the low-efficiency\naccretion state ought to show a steady jet, and they should not be\ndominated by thermal emission from the accretion disk, i.e., they\nshould not have a ``big blue bump''.\n\nThese objects are all low-luminosity AGN.  Most are classified as\nSeyfert galaxies (including both Seyfert 1, 2, and intermediate types)\nor low-ionization nuclear emission-line region \\citep[LINER;\n  see][e.g.]{Hec80}.  Some of the narrow-line objects show evidence\nfor hidden broad-line regions in polarized light. Mrk~3 and NGC~1052,\n2110, and 4258 show clear radio jets \\citep{LiuZhang02}, while the\nevidence for a radio jet in NGC 2841 is only tentative so far\n\\citep{NFWea02}.  Thus, about one third of the objects have clear\ndetections of radio jets. However, the lack of clear jet detections in\nthe remainder of the objects does not necessarily imply the absence of\njets, but may be due to the lack of suitable observations.  Indeed,\namong the \\emph{hard}-state galactic BHXBs, only Cygnus X-1 has an\nimaged jet, while the presence of jets in other objects is inferred by\nindirect arguments, typically lower limits on the source size inferred\nfrom the radio brightness temperature.\n\n\\citet{Ho99} considers the SEDs of seven low-luminosity AGN, showing\nthat the ``big blue bump'' is weak or absent in these. As those\nobjects are similar to the low-\\ensuremath{\\dot m}\\ objects identified here, it is\nreasonable to expect that the SEDs are also similar, i.e., that the\n``big blue bump'' is absent and the objects have a predominantly\nnon-thermal spectrum, as expected in the analogy to black-hole\nbinaries.  However, \\citet*{SSD04} argue that the lack of an\nobservable ``big blue bump'' is better explained by absorption\n\\citep[this possibility had also been discussed by][]{Ho99}.  In\nparticular, \\citet{SSD04} claim that the observed infrared emission\nlines and X-ray luminosities argue in favor of the absorption\nhypothesis.  They furthermore cite results of detailed photoionization\nmodeling which favor a black-body ionizing continuum.  Thus, the\nevidence for an intrinsic absence of a ``big blue bump'' in\nlow-luminosity AGN as a class is inconclusive at present.\n\nThe spectral energy distributions (SEDs) of two of the\nlow-\\ensuremath{\\dot m}\\ objects identified here have been analyzed in detail:\n\\citet{LACea96} fitted the SED of NGC~4258 with an ADAF spectrum;\nhowever, \\citet{FB99} argue that the SED is equally well explained by\njet emission and an ADAF is not necessary, while \\citet{YMFea02} argue\non the basis of more recent infrared data that a jet dominates the\nemission, while an ADAF still contributes.  \\citet*{AUH04} present a\nsimilar juxtaposition of ADAF and jet models for the radio emission in\nsix other low-luminosity AGN, which have a spectral shape similar to\nADAF models but too high a radio luminosity.  There is a similar\ncontroversy for NGC~1052, for which \\citet{GOOea00} propose an ADAF\nmodel, while more recent Chandra observations presented by\n\\citet{KKRea04} support a jet model.  Regarding the shape of the\noptical continuum in this object, \\citet{KM83} claim that the narrow\nemission-line ratios are best explained with a simple power-law\nionizing spectrum, while \\citet{Peq84} argues that a black-body\nspectrum with a higher temperature than is usual in AGN may also\naccount for the observed line ratios. In other words, the same\nemission-line ratios can be accounted for by ionizing spectra both\nwith and without a thermal component.  A similar result is reported by\n\\citet*{MVG04} who show that different assumptions about the\nhomogeneity of the narrow-line region can change conclusions from\nphotoionization modeling about the shape of the ionizing continuum\ndrastically.  Again, the evidence for the absence of a ``big blue\nbump'' is inconclusive. More detailed multiwavelength observations and\ncomparison to photoionization models are necessary.\n\n\\citet{HP01} compute radio-to-optical ratios for low-luminosity AGN\nusing only optical emission from the nucleus itself (the larger\napertures used in earlier measurements included a large contribution\nfrom starlight).  The resulting radio-to-optical ratios are extremely\nlarge (up to $10^4$), similar to or even exceeding those of radio\ngalaxies and quasars with powerful jets. This suggests that radio jets\nare common in low-luminosity AGN, as predicted by the analogy with\nblack-hole binaries.  However, while this analogy seems to predict\n\\emph{both} an SED without strong thermal emission \\emph{and} the\npresence of jets, observations such as those by \\citet{AUH04} and\n\\citet{KKRea04} seem to imply that these are mutually exclusive when\nattempting to account for the radio or X-ray emission of\nlow-luminosity AGN.  \n\nThus, it is at present unclear whether low-luminosity AGN such as the\nlow-accretion rate objects identified in Fig.~\\ref{f:obs} actually are\nscaled versions of black-hole binaries in the \\emph{steady-jet, hard\n  X-ray} state.  The best evidence in favor of this hypothesis seem to\nbe the common scaling relations between X-ray binaries in the\n\\emph{steady-jet, hard X-ray} state and AGN accreting at low Eddington\nratios, as reported by \\citet*{MHD03} and \\citet*{FKM04}.  However,\nthe same scaling relation seems to extend to objects at larger\nEddington ratios as well, only with larger scatter; see Fig.~7 in\n\\citet{MHD03}. Even a rebinned version of that figure, Fig.~2 in\n\\citet{MGF03}, shows \\emph{steep power-law} (or \\emph{very high\n  state}) BHXBs and FR~II radio galaxies falling on the same relation\nas the low-efficiency sources.  It is somewhat surprising that the\nscaling does not break down altogether for the presumed\nhigh-efficiency and super-Eddington sources sources, given that the\ninnermost regions of the accretion disks are supposed to be in\nentirely different physical states in those objects.\n\n\\subsection{Spectral evidence for efficient accretion?}\n\\label{s:disc.spec.higheff}\n\nAs a corollary to the situation for low-efficiency accretors, AGN in\nthe high-efficiency state should be analogues of black-hole binaries\nin the \\emph{thermal-dominant} state.  When an object enters the\n\\emph{thermal-dominant} state, the radio emission associated with the\nsteady jet is quenched \\citep{TGKea72}, indicating that there is a\nmuch weaker jet, or no jet at all, in this state \\citep{GFP03}.\n\\citet{MGF03} show that the radio luminosity of AGN with Eddington\nratios placing them in the \\emph{thermal dominant} state drops below\nthe correlation found for sources in the \\emph{hard} state (the\ncorrelation is based on a binned version of the data from\n\\citealp{MHD03} and similar to that found by \\citealp{FKM04}), which\nis evidence in favor of such an analogy.  A similar suggestion\nhas been made by \\citet{GC01}, who show that the line separating\nlow-power FR~I and high-power FR~II sources in a plot of AGN radio\nluminosity against host galaxy optical luminosity can be interpreted\nas a threshold in accretion rate separating the two populations.\nHowever, \\citet{CR04} show that at least one-third of the FR~I sources\nin the 3C catalog must be accreting in the radiatively efficient\nregime to account for their radio luminosities, if their jets are to\nbe powered by the Blandford-Znajek mechanism.  As mentioned above,\nthey ascribe the low optical core flux of these sources to absorption.\n\nWe can assess whether the presence of jets is restricted to objects in\nthe inefficient regime by considering Figure~\\ref{f:3cobs} again. What\nmatters to the analogy between black-hole binaries and AGN is that\n\\emph{all} objects plotted in Figure~\\ref{f:3cobs} show extended radio\nemission powered by jets.  In the BHXB-AGN unification picture, none\nof the objects with $\\ensuremath{\\dot m}>\\ensuremath{\\dot m_{\\mathrm{crit}}}$ ought to show steady jets. Of the\nradio-loud quasars and radio galaxies from the 3C survey, at least the\nradio-loud quasars are obviously not in the inefficiently accreting\nregime in Figure~\\ref{f:3cobs}.  These objects are the most luminous\nradio sources in the universe. As jet kinetic power correlates nearly\nlinearly with low-frequency radio power \\citep{Wileta99}, these\nsources also have the most powerful jets in the universe. This is\ninconsistent with the expectation that the jets should be quenched\n\\citep[even though the quenching appears to be less extreme in AGN\n  than in BHXB, the AGN quenching may have been underestimated due to\n  systematic errors in luminosity and black hole\n  measurements;][]{MGF03}.  I now discuss a possible resolution of\nthis contradiction.\n\n\\subsection{Are some radio-loud AGN in a ``steep power-law'' state?}\n\\label{s:disc.thirdstate}\n\nDisks in efficiently accreting AGN do appear to be able to launch\nrelatively more powerful jets than BHXBs with identical $\\ensuremath{\\dot m}$,\napparently implying that BHXB-AGN unification is not perfect.\nHowever, the suggestion has been made that the radio galaxies and\nquasars with powerful jets are analogues of black-hole binaries in the\n\\emph{steep power-law} state with transient X-ray and radio flares,\ninterpreted as discrete ejections of high-velocity jets\n\\citep{GFP03,Mei01}. The observations that have most specifically been\ntaken as evidence for this hypothesis are dips in the X-ray emission\nfrom the blazar 3C\\,120 linked to the appearance of new radio knots in\nits jet \\citep{MJGea04}.  A naive scaling of the duration of these\noutburst in stellar-mass systems (few days) by a factor of the black\nhole mass ratio for AGN ($10^8$--$10^{10}$) results in expected jet\nlifetimes of order $10^6$\\,y -- $10^8$\\,y for these AGN. It is\nencouraging that this brackets the typical jet age of powerful radio\nsources of $\\approx 10^7$\\,y \\citep{Wileta99}.  However, all the\nradio-loud quasars in Figure~\\ref{f:3cobs} have $\\ensuremath{\\dot m} \\approx 0.1$,\ni.e., in the standard thin-disk jetless regime.  Alternatively, if the\naccretion rates of the radio-loud quasars are systematically\nunderestimated by a factor of 2--3, these sources may be in the regime\nof the \\emph{steep power-law} state (indeed, the black hole masses of\nsome of the high-redshift 3CR radio-loud quasars may have been\nunderestimated by a comparable factor, see the note at the end of\n\\S\\ref{s:obs.err}).  But even if the accretion rates derived for the\nradio-loud quasars are not correct, there are other sources with\npowerful jets, but implied accretion rates firmly in the standard-disk\nregime.  This suggests that additional factors besides the accretion\nrate govern the disk structure and the appearance of jets (as proposed\nby \\citealp{Mei01}; also compare \\citealp*{CF00,LPK03}).  This is\nfurther supported by the detection of extended radio emission with\nFR~I morphology around the quasar E1821+643, which is optically\nluminous but has very low radio luminosity \\citep{BR01}.  Conventional\nwisdom has it that optically luminous quasars either have FR~II jets\n(if they are ``radio loud'' by either criterion for radio loudness),\nor a jet that is so weak that it cannot leave the host galaxy (or no\njet at all).  While this particular object may have FR~I morphology\nbecause of strong jet precession \\citep{BR01}, it is possible that\nthere are many more optically luminous quasars whose extended emission\nhas not been detected because of a lack of sufficiently deep radio\nobservations.\n\nThe ``fast ejection'' scenario for jets in powerful sources makes\ntestable predictions about the X-ray spectra of FR~I and FR~II radio\nsources. Radio-loud AGN typically have a harder X-ray spectrum than\nAGN with similar optical, but lower radio luminosity, i.e., weaker or\nabsent jets \\citep{EWMea94}. This matches the spectrum of BHXBs showing\nhigh-speed ejections in the \\emph{very high} state, whose distinctive\nfeature is in fact a \\emph{steep X-ray power-law} \\citep[SPL;\nsee][]{MR04}, i.e., an X-ray spectrum that is softer than that in the\n\\emph{jet\/hard} state (but still harder than in the \\emph{thermal dominant}\nstate).  As these ejections are non-steady, there are a large number\nof BHXBs in the SPL state but without strong radio emission.  The\nunification scheme thus predicts the presence of AGN with hard X-ray\nspectra similar to those of radio-loud quasars, but without powerful\njets.\n\nThere is a possible exception to the radio-loud AGN unification scheme\nwith implications for BHXB-AGN unification. \\citet{EH03} argue that\nbroad-line radio galaxies (BLRGs) with double-peaked emission lines\naccrete in the low-efficiency regime.  Their optical luminosity would\nthen be lower compared to quasars with similar radio luminosity not\nbecause of obscuration, but because of a flow with lower radiative\nefficiency.  In this case, the radio galaxies with double-peaked lines\nare relatives of \\emph{jet\/hard}-state BHXBs with steady, low-speed\njets, not of their fellow FR~II radio sources with optical quasar\nspectra and presumed high-speed non-steady ejections (jets).  Indeed,\nBLRGs with double-peaked lines have narrow-line ratios indicating\nlower ionization states than is typical for radio-loud objects.  \n\nThis separation into high- and low-ionization jet sources matches the\nfindings by \\citet{WRBea01} who show that is is necessary to invoke a\ndual-population scheme for radio galaxies in order to fit the\nevolution of the radio luminosity function.  The high-luminosity\npopulation includes all FR~II sources with strong emission lines,\nwhile the low-luminosity population includes all FR~I and those FR~II\nsources with weak emission lines, and presumably those BLRG with\ndouble-peaked emission lines.  Based on such dual-population schemes,\n\\citet{Mei01} had already proposed that there are two subclasses of\njetted AGN with different modes of accretion and jet generation.\nThus, both the double-peaked BLRGs and the dual-population scheme\nrepresent independent evidence for a class of FR~II radio galaxies\nwith different nuclear properties that can be explained by\nlower-efficiency accretion.  It is important, though, that the\nluminosity separating the two radio populations with different\nevolution is about a factor of 10 higher than the luminosity\nseparating sources with FR~I and FR~II morphology, i.e., this\nseparation is \\emph{not} the one considered by \\citet{GC01}.\n\nIf the BHXB-AGN analogy holds up, there should be detectable\ndifferences between the jets in the high- and low-luminosity radio\npopulations in terms of speeds and lifetimes.  In this context, the\nstudy of radio galaxies with two sets of radio lobes (double-double\nradio galaxies) may become important.  These objects are interpreted\nas AGN whose activity has been interrupted for a few Myr\n\\citep{SBRea00} and their study may provide insights into jet\ntriggering mechanisms.\n\nAnother prediction of the BHXB-AGN unification including the third\naccretion sate is that there should be AGN equivalents of BHXBs in the\nSPL state that do not currently produce ejections, i.e., AGN which are\nidentical to FR~II radio galaxies and quasars except for emission that\ncan be ascribed to jets, but which are different from jetless AGN in\nthe \\emph{thermal-dominant} state. These issues deserve further\ninvestigation that is, however, beyond the scope of the present work.\n\n\n\\section{Summary and conclusion}\n\\label{s:conc}\n\nEvidence is accumulating now that the different accretion states\nobserved in black-hole binaries \\citep{MR04} will analogously occur in\naccretion systems in active galactic nuclei, i.e, for a unification of\nblack-hole binaries (BHXB) and active galactic nuclei (AGN)\n\\citep{Mei01,MGF03}.  In this paper, I present a simple observational\ntest for the existence in AGN of the transition between a radiatively\ninefficient state and an efficient one thought to occur at a critical\nEddington-scaled accretion rate $\\ensuremath{\\dot m_{\\mathrm{crit}}} \\approx 0.01$ in BHXBs\n\\citep{Mac03}.  The test is based on the determining the location of\nobjects in the $(\\Mbh,\\Lbol)$ plane, and hence their Eddington-scaled accretion\nrate.  The underlying distribution of black hole masses and physical\naccretion rates is expected to vary smoothly. Therefore, a change in\nthe accretion efficiency implies that the distribution of objects in\nthe $(\\Mbh,\\Lbol)$ plane\\ should also change at the critical accretion rate.  I\nconsider whether such a change is detectable in a sample of AGN with\nknown bolometric luminosities and black hole masses obtained from\n\\citet{WU02a} and \\citet{MD04}.  I obtain the following results:\n\\begin{enumerate}\n\\item The bulk of objects considered here lie in the radiatively\n  efficient regime $\\ensuremath{\\dot m} > \\ensuremath{\\dot m_{\\mathrm{crit}}}=0.01$ of the\n  $(\\Mbh,\\Lbol)$ plane\\ (Figure~\\ref{f:obs}).  Even though the lack of objects at\n  $\\ensuremath{\\dot m} < \\ensuremath{\\dot m_{\\mathrm{crit}}}$ with $\\ensuremath{M_{\\mathrm{BH}}} > 10^9\\,\\ensuremath{M_{\\sun}}$ is mainly due to\n  selection effects, the small number of objects with $\\ensuremath{M_{\\mathrm{BH}}} <\n  10^9\\,\\ensuremath{M_{\\sun}}$ and $\\ensuremath{\\dot m} < \\ensuremath{\\dot m_{\\mathrm{crit}}}$ is weak suggestive evidence that\n  the density of objects decreases below the line $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}$.\n\\item Selection effects are important in shaping the observed\n  distribution of objects in the $(\\Mbh,\\Lbol)$ plane, in particular those\n  selection effects arising in flux-limited samples\n  (\\S\\ref{s:obs.selection}).  The most severe bias is the lack of\n  low-luminosity objects at high redshifts from existing black hole\n  mass surveys.  These objects outnumber luminous AGN at all redshifts\n  and likely have a different distribution of accretion rates than\n  low-luminosity objects in the local universe.  In particular, the\n  apparent bimodality of the accretion rate distribution of AGN with\n  powerful jets \\citep{MCF04} is predominantly due to this selection\n  against low-luminosity high-redshift AGN.  The bimodality is\n  strongly reduced if high-accretion rate objects are restricted to\n  lie at $z<0.29$, the maximum redshift of the low-accretion rate\n  objects in the \\citet{MCF04} sample.\n\\item Without a reliable method to determine an individual black\n  hole's spin and hence the fraction $\\eta$ of binding energy\n  liberated by accretion, all determinations of accretion rates may be\n  uncertain by a factor of three or greater (from $\\eta=0.42$ for\n  maximally co-rotating black holes to $\\eta=0.06$ for non-rotating\n  black holes, or even less for counter-rotating black holes; see\n  \\S\\ref{s:obs.assumptions}).  This and similar uncertainties in the\n  determination of black hole masses and bolometric luminosities\n  mainly blur the expected change in object density at $\\ensuremath{\\dot m}=\\ensuremath{\\dot m_{\\mathrm{crit}}}$\n  (\\S\\S\\ref{s:obs.err}, \\ref{s:obs.lum}).  The systematic effects\n  associated with the use of a single-band flux and a bolometric\n  correction factor are potentially more seriously distorting the\n  apparent distribution of objects, calling for multiwavelength SEDs\n  to be determined for a larger sample of AGN.\n\\item The objects identified as low-efficiency accretors in\n  Figure~\\ref{f:obs} are all low-luminosity AGN\n  (\\S\\ref{s:disc.spec.loweff}).  The evidence for jets and against the\n  presence of thermal ``big blue bump'' emission from these objects is\n  inconclusive on close scrutiny, although low-luminosity AGN have\n  radio-to-optical ratios similar to those of powerful radio galaxies\n  and radio-loud quasars with jets if only their nuclear fluxes are\n  considered \\citep{HP01}.  Jet and ADAF models are often pitted\n  against each other in the literature to explain the radio and X-ray\n  emission from low-luminosity AGN, while the unification would\n  predict a simultaneous contribution from both \\citep[as obtained by\n    some authors, see][e.g.]{UH01,YMFea02}. Furthermore,\n  photoionization models can account for the narrow-line emission from\n  these objects by an ionizing continuum both with and without a\n  thermal component \\citep{MVG04}. Hence, the apparent lack of a big\n  blue bump does not necessarily imply the absence of thermal emission\n  from the accretion disk, but could also be due to absorption.  On\n  the other hand, broad-line radio galaxies with double-peaked\n  emission lines may be a class of objects that is genuinely in the\n  low-efficiency regime \\citep{EH03}.\n\\item The energy output of some or all low-efficiency BHXBs is\n  dominated by the kinetic energy of jets, so that observational tests\n  for an equivalent kinetic energy output from low-luminosity AGN\n  should be a powerful test of BHXB-AGN unification, e.g., by looking\n  for evidence for deposition of the jet's kinetic energy in the AGN's\n  surroundings.  \n\\item Many objects in the 3C sample have both the most powerful radio\n  known radio jets and accretion rates placing them firmly in the\n  radiatively efficient regime (Figure~\\ref{f:3cobs} and\n  \\S\\ref{s:disc.spec.higheff}), where jet production should have been\n  quenched according to the BHXB-AGN unification scheme \\citep[as\n    noted by][]{Mei01}.  These objects might be the counterparts of\n  BHXBs in the \\emph{steep power-law} (SPL) state with non-steady fast\n  jets \\citep[as suggested for 3C\\,120;][]{MJGea04}.  Double-double\n  radio galaxies, in which the jets have been interrupted for a few\n  Myr \\citep{SBRea00}, perhaps constitute further evidence for such an\n  analogy.  In these objects, jet production is obviously triggered by\n  factors other than the accretion rate. Similarly, there should be\n  AGN counterparts of SPL BHXBs which are not currently undergoing\n  ejections.\n\\end{enumerate}\n\nA more accurate determination of the distribution of objects in the\n$(\\Mbh,\\Lbol)$ plane\\ clearly depends on more accurate black hole mass\nmeasurements. The ideal sample to avoid selection effects would be a\nvolume-limited AGN catalog with complete black hole mass\nidentification.  This would allow a determination of the accretion\nrate distribution in a manner similar to the computation of luminosity\nfunctions (determining volume densities instead of counting\nincidences).  Modern surveys like the Sloan Digital Sky Survey's\nspectroscopic quasar survey find quasars with a much broader range in\noptical colors, i.e., optical SED shapes, than the Bright Quasar\nSurvey \\citep{SG83}. Therefore, more multiwavelength SEDs are needed\nto obtain an accurate measurement of \\ensuremath{L_{\\mathrm{bol}}}\\ for SDSS quasars, removing\nthe substantial uncertainty associated with extrapolation from optical\nto bolometric luminosity. Since the X-ray emission is particularly\nimportant to test the spectral equivalence of AGN and BHXB accretion\nstates, a sample of normal AGN with rest-frame hard X-ray spectra,\ndeep VLA imaging to search for extended radio emission like that\ndetected around E1821+643 \\citep{BR01}, and well-determined black hole\nmasses would be particularly valuable to confirm that quasars are the\ncounterparts of jetless BHXBs in the \\emph{thermal dominant} state.\nClearly, the simplified theoretical treatment in \\S\\ref{s:theory}\nshould be refined by using more detailed relations between the\nradiated flux in different parts of the spectrum and \\ensuremath{\\dot m}\\ for\ndifferent accretion scenarios. In particular, this will allow to\nexplore possible relations between radio loudness and accretion mode\n(see \\citealp{MGF03} and \\citealp*{WHS03}). In a broader context, the\nobserved joint distribution of luminosities and black hole masses can\nbe related to a plausible underlying distribution of accretion rates\nand resulting radiative efficiencies for specific models of accretion,\nto test their applicability to the AGN population as a whole\n\\citep{Mer04}.\n\n\\acknowledgements\n\nThis work was supported by the U.S.\\ Department of Energy under\ncontract No.\\ DE-AC02-76CH03000.  I am grateful to Ross McLure for\nproviding the luminosity and black hole mass measurements from\n\\citet{MD04} in electronic form, to Arieh K\\\"onigl for a critical\nreading of the manuscript, and to Sebastian Heinz for a useful\ndiscussion.  I am particularly grateful to the anonymous referee for\nproviding rapid reviews which have led to substantial improvements in\nthe paper.\n\nFunding for the Sloan Digital Sky Survey (SDSS) has been provided by\nthe Alfred P. Sloan Foundation, the Participating Institutions, the\nNational Aeronautics and Space Administration, the National Science\nFoundation, the U.S.\\ Department of Energy, the Japanese\nMonbukagakusho, and the Max Planck Society.\\footnote{The SDSS Web site is\n\\url{http:\/\/www.sdss.org\/}.}\n\nThe SDSS is managed by the Astrophysical Research Consortium (ARC) for\nthe Participating Institutions. The Participating Institutions are The\nUniversity of Chicago, Fermilab, the Institute for Advanced Study, the\nJapan Participation Group, The Johns Hopkins University, the Korean\nScientist Group, Los Alamos National Laboratory, the\nMax-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute\nfor Astrophysics (MPA), New Mexico State University, University of\nPittsburgh, University of Portsmouth, Princeton University, the United\nStates Naval Observatory, and the University of Washington.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor the nearest-neighbor (NN) Ising antiferromagnet on the square\nlattice in a uniform magnetic field, the low temperature ordered\nphase is separated from the paramagnetic phase by a simple, 2nd\norder phase boundary.  (Within the context of the lattice gas model\nthis system could be described as having repulsive NN-coupling and\nforming a $c(2 \\times 2)$ ordered state.)  With the addition of\nrepulsive (antiferromagnetic) next-nearest-neighbor (NNN)\ninteractions the situation becomes more complicated.  Early Monte\nCarlo simulations suggested that a single, super-antiferromagnetic,\nor $(2 \\times 1)$, phase existed, separated from the paramagnetic\nphase by a single phase boundary\\cite{Lan71,Lan80,Bin80}.  A\ndegenerate, row-shifted $(2 \\times 2)$ state was also predicted at\nzero temperature.  (See Fig.~\\ref{f1} for a schematic representation of these states.)  On the other hand, symmetry arguments based on\nLandau theory\\cite{Dom78} predict the order-disorder transitions of\n$(2\\times1)$ and $(2\\times2)$ structures belong to XY model with\ncubic anisotropy.\n\nThe original motivation of this study was to investigate the\npossibility of XY-like behavior of  the Ising spins on the square\nlattice, since there is numerical evidence\\cite{Lan83} that Ising\nantiferromaget with attractive NNN interaction on the triangular\nlattice has an XY-like intermediate state between the low\ntemperature ordered state and high temperature disordered state. In\nfact, the present model has already been studied by many authors\nusing different approaches. An early Monte Carlo study\\cite{Bin80}\ncomprehensively showed the phase diagrams for several different\ninteraction ratios (R) of NNN to NN interaction. But due to the\ndeficiencies of computer resources at that time, for the $R=1$ case,\na disordered region was missed between the two ordered phases which\nwas pointed out in a later interfacial free energy\nstudy\\cite{Slo83}. Meanwhile, transfer matrix\nstudies\\cite{Kas83,Ama83} found reentrant behavior for the\n$(2\\times1)$ transition lines. While a study using the cluster\nvariation method\\cite{Lop93,Lop94} concluded that for a range of R\n(0.5$\\sim$1.2) the system undergoes a first order transition, a\nrecent Monte Carlo study\\cite{Mal06} using a variant of the\nWang-Landau method\\cite{wls} focused on the $R=1$ case without\nexternal field and found the phase transition is of second order.\nFor external field $H=4$ (in the unit of NN interaction constant)\nthe two ordered phases, namely $(2\\times1)$ and row-shifted\n$(2\\times2)$, are degenerate at zero temperature so it is tempting\nto think that the cubic anisotropy would be zero for this field and\nthat there could be a Kosterlitz-Thouless transition.\n\nIn this paper, we carefully study the location of phase boundaries\nand the critical behavior for the case $R=1$. In Sec.~\\ref{s1} the\nmodel and relevant methods and analysis techniques are reviewed. Our\nresults are presented in Sec.~\\ref{s2}, along with finite size\nscaling analyses, and we summarize and conclude in Sec.\\ref{s3}.\n\n\\section{Model and method}\\label{s1}\n\n\\subsection{The model}\nThe Ising model with NNN interaction is described by the Hamiltonian\n\\begin{equation}\n{\\cal\nH}=J_{NN}\\sum_{<i,j>_{NN}}\\sigma_i\\sigma_j+J_{NNN}\\sum_{<i,j>_{NNN}}\\sigma_i\\sigma_j+H\\sum\n\\sigma_i,\n\\end{equation}\nwhere $\\sigma_i, \\sigma_j= \\pm1$, $J_{NN}$ and $J_{NNN}$ are NN and\nNNN interaction constants, respectively, H is an external magnetic\nfield, and the sums in the first two terms run over indicated pairs of neighbors on a\nsquare lattice with periodic boundary conditions.  Both $J_{NN}$ and $J_{NNN}$ are positive (antiferromagnetic) and the ratio $R=J_{NNN}\/J_{NN}$.\n\n\\begin{figure}\n\\includegraphics[width=3.25in]{conf.eps}\n\\caption{Schematic plots of $c(2 \\times 2)$, $(2 \\times 1)$ and\nrow-shifted $(2 \\times 2)$ ordered structures within the context of\nthe lattice gas model.  (In magnetic language, filled circles correspond to up spins and empty circles correspond to down spins.)} \\label{f1}\n\\end{figure}\n\nFor the $R=1$ case, the ground states would be the $(2\\times1)$\nstate, also known as super-antiferromagnetic state, in small\nmagnetic fields; and at higher fields it would be a  row-shifted\n$(2\\times2)$ state, which differs from the $(2\\times2)$ state in the\nsense that the antiferromagnetic chains in the former state can\nslide freely without energy cost. See Fig.~\\ref{f1}. Locally, the\nstructure may appear to be $(2\\times2)$, but for large enough\nlattices the equilibrium structure always shows row shifting. As a\nresult, such a row-shifted $(2\\times2)$ state is highly degenerate,\nand the antiferromagnetic sublattice exhibits only one dimensional\nlong range order. In terms of the sublattice magnetizations\n\\begin{equation}\nM_\\lambda = \\frac{4}{N}\\sum_{i\\in\\lambda}\\sigma_i, \\quad\n\\lambda=1,2,3,4\n\\end{equation}\nwe can define two components of the order parameter for the\n$(2\\times1)$ state\n\\begin{equation}\nM^{a}=[M_1+M_2-(M_3+M_4)]\/4,\n\\end{equation}\n\\begin{equation}\nM^{b}=[M_1+M_4-(M_2+M_3)]\/4,\n\\end{equation}\nwith a computationally convenient root-mean-square order parameter\n\\begin{equation}\nM^{rms}=\\sqrt{(M^{a})^2+(M^{b})^2}.\n\\end{equation}\nSince $M^{rms}$ would have a limiting value of $\\frac{1}{2}$ for the\nrow-shifted $(2\\times2)$ state and be zero for the disordered\nstate, it can also be used as an order parameter for the row-shifted\n$(2\\times2)$ state.\n\nOther observables, such as the finite lattice ordering\nsusceptibility $\\chi$ and fourth-order cumulant $U$, are defined in\nterms of the order parameter $M^{rms}$ as\n\\begin{equation}\n\\chi=\\frac{N}{T}\\left[<(M^{rms})^2>-<M^{rms}>^2\\right]\n\\end{equation}\n\\begin{equation}\nU=1-\\frac{<(M^{rms})^4>}{3<(M^{rms})^2>^2}\n\\end{equation}\nwhere N is the total number of spins and T is the simulation\ntemperature. In some cases, the true ordering susceptibility\n$\\chi^{+}$, which is $\\frac{N}{T}<(M^{rms})^2>$, is used to\neliminate simulation errors resulting from $<M^{rms}>$, where the\norder parameter is known to be zero for the infinite lattice.\n\n\\subsection{Simulation methods}\nFor small lattice sizes, Wang-Landau sampling~\\cite{wls} was used to obtain a quick overview of the thermodynamic behavior of our model.  A two-dimensional random walk in energy and magnetization space was performed so that the density of states $g(E,M)$ could be used to determine all thermodynamic quantities (derived from the partition function) for\nany value of temperatures and external field.  Consequently, \"freezing\" problems are avoided at extremely\nlow temperatures.  This allowed us to determine the ``interesting'' regions of field-temperature space; however, it quickly became apparent that, because of subtle finite size effects, quite large lattices would be needed.  Unfortunately, as L increases, the number of entries of histogram explodes as $L^4$ and it proved to be more efficient to use parallel tempering instead.\n\nSince a large portion of interesting phase boundary is at relatively\nlow temperatures and many local energy minima exist which makes the\nrelaxation time rather long, the parallel tempering\nmethod\\cite{Swd86,Hukushima96} is a good choice for simulating our\nmodel. The basic idea is to expand the low temperature phase space\nby introducing configurations from the high temperatures. So, many\nreplicas at different temperatures are simulated simultaneously, and\nafter every fixed number of Monte Carlo steps, a swap trial is\nperformed with a Metropolis-like probability which satisfies the\ndetailed balance condition. The transition probability from a\nconfiguration $X_m$ simulated at temperature $\\beta_m$ to a\nconfiguration $X_n$ simulated at temperature $\\beta_n$ would be\n\\begin{equation}\nW(X_m,\\beta_m|X_n,\\beta_n)=min[1,exp(-\\triangle)],\n\\end{equation}\n\\begin{equation}\n\\triangle=(\\beta_n-\\beta_m)({\\cal H}_m-{\\cal H}_n).\n\\end{equation}\nWe chose the temperatures for the replicas to be in a geometric\nprogression\\cite{Kof04}, which would make acceptance rates relatively\nconstant among neighboring temperature pairs, and the total number\nof temperatures was chosen to make the average acceptance rate\nabove $20\\%$.\n\nThe multiplicative, congruential random number generator RANECU was used \\cite{James90,Ecuyer88}, and some results were also obtained using the Mersenne Twister \\cite{twister} for comparison.  No difference was observed to within the error bars.\n\nTypically, data from $10^6$ to $10^7$ MCS  were kept for each run and 3 to 6 independent runs are taken to calculate standard statistical error bars. For parallel termpering, the swap trial was attempted after every MCS.\nIn all the plots of data and analysis shown in following sections, if error bars are not shown they are always smaller than the size of the symbols.\n\nIn general, such replica exchange not only applies to the\ntemperature set, but also can apply to any other sets of fields,\nsuch as the external magnetic field. Following the same argument,\none can get the transition probability from $\\{X_m, H_m\\}$ to\n$\\{X_n,H_n\\}$\n\\begin{equation}\nW(X_m,H_m|X_n,H_n)=min[1,exp(-\\triangle)],\n\\end{equation}\n\\begin{equation}\n\\triangle=\\beta(H_n-H_m)(M_m-M_n).\n\\end{equation}\nwhere $M_m, M_n$ are the uniform magnetizations of replica m and n,\nrespectively.\n\n\\subsection{Finite-size scaling analysis}\nTo extract critical exponents from the data, we performed\nfinite-size scaling analyses along the transition lines. Since the\nmaximum slope of the fourth-order cumulant $U$ follows\\cite{Fer91}\n\\begin{equation}\\label{eq1}\n(\\frac{dU}{dK})_{max}=a^{\\prime}L^{\\frac{1}{\\nu}}(1+b^{\\prime}L^{-\\omega}),\n\\end{equation}\nwhere $K=\\frac{J_{NN}}{k_BT}$,  the correlation length exponent\n$\\nu$ can be estimated directly.\n\nWith the exponent $\\nu$ and critical temperature $T_c$ at hand, the\ncritical exponent $\\beta$ and $\\gamma$ can be extracted from the\ndata collapsing of the finite-size scaling forms,\n\\begin{equation}\nM=L^{-\\frac{\\beta}{\\nu}}\\overline{X}(tL^{\\frac{1}{\\nu}})\n\\end{equation}\n\n\\begin{equation}\n\\chi T=L^{\\frac{\\gamma}{\\nu}}\\overline{Y}(tL^{\\frac{1}{\\nu}})\n\\end{equation}\n\n\\noindent where $t=|1-\\frac{T}{T_c}|$, and $\\overline{X}$ and $\\overline{Y}$\nare universal functions whose analytical forms are not known. One\ncan also estimate the exponent $\\alpha$ from the relation of the\npeak position with lattice size for the specific heat\n\\begin{equation}\\label{eq2}\nC_{max}=cL^{\\frac{\\alpha}{\\nu}}+C_0\n\\end{equation}\nwhere $C_0$ is the \"background\" contribution. In some cases when the\nappropriate paths, i.e. which are perpendicular to the phase\nboundary, are ones of constant temperatures, then the critical\nbehavior would be expressed in terms of reduced field\n$h=|1-\\frac{H}{H_c}|$, and all the above analysis still applies.\n\n\\subsection{GPU acceleration}\nGeneral purpose computing on graphics processing units(GPU) attracts\nsteadily increasing interest in simulational physics\\cite{Yan07,\nAnd08, Pre09}, since the computational power of recent GPU exceeds\nthat of a central processing unit(CPU) by orders of magnitude. The\nadvantage continues to grow as the performance of GPU's doubles\nevery year. Recently, a GPU accelerated Monte Carlo simulation of\nIsing models\\cite{Pre09} was performed. Compared to traditional CPU\ncalculations, the speedup was about 60 times.\n\nThe idea behind the implementation in Ref \\cite{Pre09} can be easily\nextended to our model, and the parallel tempering algorithm is\nnaturally realized. Initially, all the replica are loaded to the\nglobal memory of the GPU. For each replica, the entire lattice is\ndivided into four sublattices, then spins in the same sublattice can\nbe updated simultaneously by the GPU using a Metropolis scheme, and\nthe swap of configurations of replica pairs can also be achieved in\nparallel.\n\nOn the GeForce GTX285 graphics unit, our code runs about 10 times faster than it\ndoes on the 32 CPUs of IBM p655 cluster using Message Passing\nInterface(MPI) for parallel computation.\n\n\n\\begin{figure}\n\\includegraphics[width=0.95\\columnwidth]{pd.eps}\n\\caption{The phase diagram for the Ising square lattice with\nantiferromagnetic nearest- and next-nearest-neighbor interactions in\na magnetic field for $R=1$. Open circles and pluses denote\nsimulation results. The solid lines are second order transition\nlines, while the dashed line indicates the short range ordering\nline.} \\label{f2}\n\\end{figure}\n\n\n\\section{Results and Discussion}\\label{s2}\n\\subsection{Phase diagram and short range ordering}\nFrom the ground state analysis, with zero or low field the ordered\nstate would be the superantiferromagnetic, or $(2\\times1)$,\nstructure.  As the external field increases to $4<H\/J_{NN}<8$, more\nspins align in the opposite field direction, and the ordered\nstructure would be the row-shifted $(2\\times2)$. With even stronger\nfields, the system becomes paramagnetic.  In the region near\n$H\/J_{NN}=4$ a mixture of $(2\\times1)$ and row-shifted $(2\\times2)$\nis visible.\n\nFor finite temperatures, we found that the fourth-order cumulant is\nalways a good quantity to use to locate the transition points, while the\ndata for other quantities, such as the specific heat or susceptibility,\nmay look \"strange\" due to the effect of neighboring critical\npoints. To help the reader understand the observed thermodynamic properties, the final phase diagram for $R=1$ is plotted in Fig.~\\ref{f2}:  The\n solid lines are the phase boundaries, all of which are\ncontinuous. The dashed line inside the $(2\\times1)$ ordered phase\nindicates a ``short range ordering'' line, which was located from\nthe peak position of the specific heat. As shown in Fig.~\\ref{f3},\nfor paths of constant temperature $k_BT\/J_{NN}=1.1$ and $1.2$, there\nare two peaks, and the one that increases with lattice size\ncorresponds to the critical point.\n\n\n\\begin{figure}\n\\includegraphics[width=0.95\\columnwidth]{csro2.eps}\n\\vspace{0.5cm}\\\\\n\\includegraphics[width=0.95\\columnwidth]{csro1.eps}\n\\caption{Variation of the specific heat $C$ versus field $H$ with\nlattice sizes $L=20, 40, 80, 160, 200, 300$ for paths of constant:\n(a)$k_BT\/J_{NN}=1.2$ and (b)$k_BT\/J_{NN}=1.1$.} \\label{f3}\n\\end{figure}\n\n\nAn indication of the complexity of the finite size behavior is\nclearly seen in the bottom portion of Fig.~\\ref{f3} in which the\nsmall lattices actually have minima in the specific heat for field\nvalues that eventually show phase transitions for sufficiently large\nsystems.  The round-shaped size-independent peak is due to the short\nrange ordering of the $(2\\times1)$ ``clusters'' of different\norientation from the ordered background. No corresponding\nbehavior was observed from the susceptibility or the fourth-order\ncumulant.\n\nIn order to confirm the above argument, we also calculated the NN\nand NNN pair correlation function, that is $<\\sigma_i\\sigma_j>$, for\npaths of different fields crossing this line. The correlation\nfunction data are plotted in Fig.~\\ref{f4}. The NN pair correlation\ndecreases from zero to a minimum and then increases to positive\nvalues, while the NNN pair correlation increases monotonically from\n$-1$.\n\n\n\\begin{figure}\n\\includegraphics[width=0.95\\columnwidth]{nn.eps}\n\\vspace{0.5cm}\\\\\n\\includegraphics[width=0.95\\columnwidth]{nnn.eps}\n\\caption{Nearest- and next-nearest-neighbor correlation functions.\nThe field is varied for paths of 3 different temperatures:\n$k_BT\/J_{NN}=0.4,0.5$ and 0.6 across the short range ordering line.}\n\\label{f4}\n\\end{figure}\n\n\n\\subsection{Critical behavior}\nThe data for the specific heat and susceptibility for three different\nvalues of $H$ are plotted in Fig.~\\ref{f5}.\n\n\n\\begin{figure}\n\\includegraphics[width=0.95\\columnwidth]{c.eps}\n\\includegraphics[width=0.95\\columnwidth]{sus.eps}\n\\caption{Specific heat and susceptibility for 3 different fields\n across the phase boundary. Data are for:\nL=100, $\\vartriangle$; L=120, $\\bullet$; L=160, $\\times$; L=200\n$\\blacktriangle$; L=300 $+$; L=400 $\\ast$.} \\label{f5}\n\\end{figure}\n\n\nWithout the field, they\nboth show very sharp peaks, and from the magnitudes of the peak values of the specific\nheat, as shown in Fig.\\ref{f6}(b), we can obtain a rather accurate\nestimate of the exponent ratio $\\alpha\/\\nu=0.357(8)$, which is\nobviously not zero. In Fig.\\ref{f6}(a), we also show the curve-fit\nfor the maximum slope of $\\frac{dU}{dK}$ for $H=0$, and extract\nthe exponent $\\nu=0.847(4)$ directly.\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{dudt.eps}\n\\vspace{0.5cm}\\\\\n\\includegraphics[width=0.9\\columnwidth]{cmax.eps}\n\\caption{Curve fits using the leading terms of equations \\ref{eq1}\nand \\ref{eq2} for the maximum slopes of $\\frac{dU}{dK}$ (a) and peak\nvalues of specific heat (b), respectively.} \\label{f6}\n\\end{figure}\n\n\nBoth values of $\\alpha$ and\n$\\nu$ are quite consistent with the early estimates in\nref~\\cite{Bin80}, and the value of $\\alpha\/\\nu$ is different from\nref~\\cite{Mal06}, in which the $1\/L$ correction term was assumed up\nto lattice size $L=160$.\n\nThe same procedure was repeated for $H\/J_{NN}=2.5$ and $3.3$,\nhowever, as shown in Fig.~\\ref{f5}, the peaks of the specific\nheat become increasingly rounded as the field increases, which makes it rather\ndifficult to get a direct estimate of the exponent $\\alpha$.  Because of this it was necessary to obtain data for much larger lattice sizes, a task that was only possible with the use of GPU computing. In\nfact, as the value of $\\nu$ increases with the field, for\n$H\/J_{NN}=3.3$, according the hyper-scaling law $\\alpha=2-d\\nu$,\nwhere $d=2$ is the dimension of the system, $\\alpha$ should be\nnegative. Although the curve-fit is not stable, given the value of\n$\\alpha$ we can get a fit within error bars. Such continuous\nincreasing of the exponent $\\nu$ up to values much greater than one\nis actually consistent with the findings of an early transfer-matrix\nstudy\\cite{Ama83}.\n\nTo estimate the critical exponents $\\beta$ and $\\gamma$, we performed data\ncollapsing with a large range of lattice sizes for the order\nparameter and its susceptibility. As shown in Fig.~\\ref{f7}, the data in both finite size scaling plots\ncollapse very nicely onto single curves, and the ratio $\\beta\/\\nu$ and $\\gamma\/\\nu$\n agree with values of the 2D Ising universality class within error bars.\n\n\n\\begin{figure}\n\\vspace{0.3cm}\n\\includegraphics[width=1.0\\columnwidth]{mclps.eps}\n\\vspace{0.5cm}\\\\\n\\includegraphics[width=1.0\\columnwidth]{xclps.eps}\n \\caption{Finite size scaling data collapsing along\npaths of constant $H\/J_{NN}=0$ and $2.5$ for root-mean-square order\nparameter and its ordering susceptibility, respectively.\n$t^{\\prime}=|1-\\frac{T_c}{T}|$ and $t=(1-\\frac{T}{Tc})$. Data are\nfor: L=80, $\\circ$; L=100, $\\vartriangle$; L=120, $\\bullet$; L=160,\n$\\times$; L=200 $\\blacktriangle$; L=300, $+$; L=400, $\\ast$.}\n\\label{f7}\n\\end{figure}\n\n\nHence, although the individual exponents are non-universal, Suzuki's\nweak universality holds quite well. Another data collapsing along\nthe path of constant $H\/J_{NN}=6$ across the phase boundary of the\nrow-shifted $2\\times2$ state is shown in Fig.~\\ref{f8}.  The quality\nof the data collapsing is also excellent, and again, the exponents\nare non-universal.  The estimate for $\\beta\/\\nu$ is a bit low but\n$\\gamma\/\\nu$ agrees well with prediction of weak universality.\n\n\n\\begin{figure}\n\\vspace{0.3cm}\n\\includegraphics[width=1.0\\columnwidth]{h6mclps.eps}\n\\vspace{0.5cm}\\\\\n\\includegraphics[width=1.0\\columnwidth]{h6xclps.eps}\n\\caption{Data collapsing along the path of constant $H\/J_{NN}=6$ for\nroot-mean-square order parameter and its ordering susceptibility,\nrespectively. Data are for:  L=80, $\\circ$; L=100, $\\vartriangle$;\nL=120, $\\bullet$; L=160, $\\times$; L=200 $\\blacktriangle$; L=300,\n$+$; L=400, $\\ast$.} \\label{f8}\n\\end{figure}\n\n\n\n\n In Table.~\\ref{t1}, the critical points and exponents $\\alpha,\n\\beta, \\nu$ and $\\gamma$ for several typical paths of constant H or T\nacross the phase boundary are listed.\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{cclllll} \\hline \\hline\n   \\multicolumn{1}{c}{$order$} &\\multicolumn{1}{c}{$path $}   & \\multicolumn{1}{c}{$T_c$ or $H_c$}\n              & \\multicolumn{1}{c}{$\\alpha$}   & \\multicolumn{1}{c}{$\\beta$}\n              & \\multicolumn{1}{c}{$\\gamma$}   & \\multicolumn{1}{c}{$\\nu$} \\\\ \\hline\n                                            &{H=0}&{$2.0820(4)$}   & {$0.302(7)$} & {$0.103(3)$} & {$1.482(7)$}  & {$0.847(4)$}    \\\\\n         {$2\\times1$}                       &{H=2.5}&{$1.6852(3)$}  & {$0.104(19)$} & {$0.118(3)$} & {$1.657(6)$}  & {$0.947(7)$}    \\\\\n                                            &{H=3.3}&{$1.3335(6)$}   & {} & {$0.130(5)$} & {$1.930(6)$}  & {$1.102(8)$}\n   \\\\ \\hline\n                                            &{H=6}&{$0.7293(7)$}   & {} & {$0.110(5)$} & {$2.072(6)$}  & {$1.176(9)$}    \\\\\n   \\raisebox{2.5ex}[0pt]{$2\\times2$*}        &{T=0.5}&{$6.848(5)$}   & {} & {$0.126(4)$} & {$1.775(5)$}  & {$1.02(2)$}    \\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\caption{ Values of critical point temperatures or magnetic fields and corresponding critical\nexponents for several paths of constant temperature or field across the phase boundary\nof the $(2\\times1)$ and *row-shifted $(2\\times2)$ ordered\nphases.}\\label{t1}\n\\end{table}\n\n\n\\subsection{Reentrance behavior}\nClose to the region between the two ordered phases the correlation\nlength exponent $\\nu$ turns out to be quite large, and\ncorrespondingly the location of the critical points becomes very\ndifficult to determine. In addition, the specific heat curves look\n\"strange\", see Fig.~\\ref{f3}, because the exponent $\\alpha\/\\nu$\nwould have a negative value with large magnitude, much larger\nlattice size is needed to approach to the limiting peak value. Since\nSuzuki's weak universality seems to hold along the transition line,\nwe fixed the values of $\\beta\/\\nu=0.125$ and $\\gamma\/\\nu=1.75$ for\nthe data collapsing analysis to get a better estimate of the\ncritical point and the exponent $\\nu$.\n\n\n\n\\begin{figure}\n\\vspace{0.3cm}\n\\includegraphics[width=1.0\\columnwidth]{t0.7bc.eps}\n\\caption{Fourth order cumulant $U$ versus field $H$ along the path\nof constant $k_BT\/J_{NN}=0.7$ for lattice size $L=32,64,128,256$ .}\n\\label{f9}\n\\end{figure}\n\n\\begin{figure}\n\\vspace{0.3cm}\n\\includegraphics[width=1.0\\columnwidth]{t0.7mclps.eps}\n\\vspace{0.5cm}\\\\\n\\includegraphics[width=1.0\\columnwidth]{t0.7xclps.eps}\n \\caption{ Finite size scaling data collapsing along\nthe path of constant $k_BT\/J_{NN}=0.7$ for root-mean-square order\nparameter and its susceptibility, respectively.\n$h^{\\prime}=|1-\\frac{H_c}{H}|$ and $h=(1-\\frac{H}{Hc})$. Data are\nfor: L=32, $\\triangleleft$; L=64, $\\blacktriangleright $; L=128,\n$\\triangleright$; L=256, $\\blacktriangleleft$.} \\label{f10}\n\\end{figure}\n\n\n\nAs shown in Fig.~\\ref{f9}, the crossing point of the fourth order\ncumulant curves for a path of constant $k_BT\/J_{NN}=0.7$ is slightly\nabove $H\/J_{NN}=4$, and from the data collapsing analysis, see\nFig.~\\ref{f10}, we obtained an estimate of the critical point to be\n$H_c\/J_{NN}=4.052(7)$.   Hence, we confirm the reentrant behavior of\nthe $(2\\times1)$ transition line.\n\nFor the paths of constant $k_BT\/J_{NN}=0.45$, the curves of the\nfourth order cumulant shows two crossing points and the finite size\neffect is quite obvious. See Fig.~\\ref{f11}.\n\n\n\n\\begin{figure}\n\\vspace{0.3cm}\n\\includegraphics[width=1.0\\columnwidth]{bct0.45.eps}\n\\caption{Fourth order cumulant $U$ versus field $H$ along the path\nof constant $k_BT\/J_{NN}=0.45$ for lattice size $L=64,128,256$.}\n\\label{f11}\n\\end{figure}\n\n\nFor the larger lattice size, the two crossing points move towards\nlower fields but they do not approach each other.  Thus, a region of\ndisorder remains between the two different ordered states, even down\nto quite low temperatures.  (If however, small lattices are used\nwith insufficient data  precision, it looks as though the curves for\ndifferent lattice sizes coincide.  Such behavior would indicate,\nerroneously, the presence of an XY-like region.)  In Fig.~\\ref{f12},\nwe show data collapsing fits along the path of constant\n$k_BT\/J_{NN}=0.5$ (and excellent data collapsing is also found along\nthe path of constant $k_BT\/J_{NN}=0.3$), which confirms that there\nis a disordered region in between the two ordered states.\n\n\n\\begin{figure}\n\\includegraphics[width=0.95\\columnwidth]{t0.5mclps.eps}\n\\vspace{0.5cm}\\\\\n\\includegraphics[width=0.95\\columnwidth]{t0.5xclps.eps}\n\\caption{Finite size scaling data collapsing along paths of constant\n$k_BT\/J_{NN}=0.5$ for root-mean-square order parameter and its\nsusceptibility, respectively. Data are for: L=80, $\\circ$; L=100,\n$\\vartriangle$; L=120, $\\bullet$; L=160, $\\times$; L=200\n$\\blacktriangle$.} \\label{f12}\n\\end{figure}\n\n\nWe thus conclude that there is no XY-like region and that the two\nphase boundaries probably only come together at a bicritical point\nat $T=0$, although we cannot exclude the possibility of a bicritical\npoint at very low, but non-zero, temperature.  However, the data do\nnot yield any hint of such a bicritical point; but the lack of data\npoints for $T < 0.2$ in Fig.2 precludes us from making a definitive\nstatement about this issue.  (Moreover, the reentrant behavior of\nthe $(2\\times1)$ phase makes it very difficult to study the approach\nto $T=0$.) The variation of the critical exponents is consistent\nwith the changing magnetic field producing different effective\nanisotropies which, in turn, is expected to yield non-universal\nbehavior\\cite{Hu87}.  Due to the large values of $\\nu$ near the\nbicritical point (and correspondingly strongly negative values of\n$\\alpha$), we consider it also extremely unlikely that tricritical\npoints could be found along these transition lines as $T$ becomes\nsmall.\n\n\n\\section{Conclusion}\\label{s3}\nWe have carried out large-scale Monte Carlo simulations for the\nsquare-lattice Ising model with repulsive (antiferromagnetic)\nnearest- and next-nearest-neighbor interactions. From the finite\nsize scaling analysis, both transitions from $(2\\times1)$ and\nrow-shifted $(2\\times2)$ ordered states to disordered states turn\nout to be continuous and non-universal. The reentrance behavior of\nthe $(2\\times1)$ transition line is confirmed, and the proximity to\nthe transition line to the $(2\\times2)$ state makes it difficult to\nuntangle the low temperature behavior unless quite large lattices\nare used.  It was only possible to obtain the precise, large lattice\ndata needed though the use of GPU computing.  No evidence for\nXY-like behavior is found, and we conclude that there is probably a\nzero temperature bicritical point.  Although the exponent $\\nu$\nvaries along the transition line, the exponent ratio $\\beta\/\\nu$ and\n$\\gamma\/\\nu$ seem to agree with that of the 2D Ising universality\nclass.\\\\\n\n{\\bf Acknowledgements:} We would like to thank K. Binder and S.-H. Tsai for\nilluminating discussions and comments and also T. Preis for introducing us to the use\nof GPU's for our calculations.  Numerical computations\nhave been performed partly at the Research Computing Center of the\nUniversity of Georgia. This work was supported by NSF grant\nDMR-0810223.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}